Symplectically knotted codimension-zero embeddings of domains in R^4
aa r X i v : . [ m a t h . S G ] A ug SYMPLECTICALLY KNOTTED CODIMENSION-ZERO EMBEDDINGS OFDOMAINS IN R JEAN GUTT AND MICHAEL USHERA
BSTRACT . We show that many toric domains X in R admit symplectic em-beddings φ into dilates of themselves which are knotted in the strong sensethat there is no symplectomorphism of the target that takes φ ( X ) to X . Forinstance X can be taken equal to a polydisk P (
1, 1 ) , or to any convex toric do-main that both is contained in P (
1, 1 ) and properly contains a ball B ( ) ; bycontrast a result of McDuff shows that B ( ) (or indeed any four-dimensionalellipsoid) cannot have this property. The embeddings are constructed basedon recent advances on symplectic embeddings of ellipsoids, though in somecases a more elementary construction is possible. The fact that the embed-dings are knotted is proven using filtered positive S -equivariant symplectichomology.
1. I
NTRODUCTION
Recent years have seen a significant improvement in our understanding ofwhen one region in R symplectically embeds into another, see e.g. [ M09 ] , [ MS12 ] , [ C14 ] . Complementing this existence question, one can ask whetherembeddings are unique up to an appropriate notion of equivalence; in partic-ular, if A ⊂ U ⊂ R this entails asking whether every symplectic embedding A , → U is equivalent to the inclusion. Somewhat less is known about thisuniqueness question, though there are positive results in [ M09 ] , [ C14 ] and neg-ative results in [ FHW94 ] , [ H13 ] . We show in this paper that modern techniquesof constructing symplectic embeddings B , → U often give rise, when restrictedto certain subsets A ⊂ B ∩ U , to embeddings A , → U that are distinct from theinclusion in a strong sense.The subsets of R (and in some cases more generally in R n ∼ = C n ) that weconsider are toric domains; let us set up some notation and recall basic defini-tions.Define µ : C n → [ ∞ ) n by µ ( z , . . . , z n ) = ( π | z | , . . . , π | z n | ) .A toric domain is by definition a set of the form X Ω = µ − ( Ω ) where Ω is adomain in [ ∞ ) n . Throughout the paper the term “domain” will always referto the closure of a bounded open subset of R n or C n ; in particular domains areby definition compact.Given Ω ⊂ [ ∞ ) n , we define(1.1) b Ω = (cid:8) ( x , . . . , x n ) ∈ R n (cid:12)(cid:12) ( | x | , . . . , | x n | ) ∈ Ω (cid:9) . Symplectic embedding problems for toric domains are currently best under-stood when the domains are concave or convex according to the following def-initions, which follow [ GH17 ] . Definition . A convex toric domain is a toric domain X Ω such that b Ω is aconvex domain in R n . Definition . A concave toric domain is a toric domain X Ω where Ω ⊂ [ ∞ ) n is a domain and [ ∞ ) n \ Ω is convex. Example 1.3. If n =
2, a convex or concave toric domain X Ω arises froma “region under a graph” Ω = { ( x , y ) | ≤ x ≤ a , 0 ≤ y ≤ f ( x ) } where f : [ a ] → [ ∞ ) is a monotone decreasing function. The correspondingtoric domain X Ω is convex if f is concave, and is concave if f is convex and f ( a ) = Example 1.4. If a , . . . , a n >
0, the ellipsoid E ( a , . . . , a n ) is defined as X Ω where Ω = (cid:8) ( x , . . . , x n ) ∈ [ ∞ ) n | P ni = x i a i ≤ (cid:9) . As a special case, the ball of capac-ity a is B n ( a ) = E ( a , . . . , a ) . Note that E ( a , . . . , a n ) is both a concave toricdomain and a convex toric domain. We will occasionally find it convenient toextend this to the case that some a i = E ( . . . , 0, . . . ) = ∅ . Example 1.5. If a , . . . , a n >
0, the polydisk P ( a , . . . , a n ) is defined as X Ω where Ω = { ( x , . . . , x n ) ∈ [ ∞ ) n | ( ∀ i )( ≤ x i ≤ a i ) } . Equivalently, P ( a , . . . , a n ) = B ( a ) × · · · × B ( a n ) . Polydisks are convex toric domains.We use the following standard notational convention: Definition . If A ⊂ C n and α >
0, we define α A = {p α a | a ∈ A } .(The square root ensures that any capacity c will obey c ( α A ) = α c ( A ) , andalso that we have E ( α a , . . . , α a n ) = α E ( a , . . . , a n ) and similarly for polydisks.)For any subset B ⊂ C n let B ◦ denote the interior of B . This paper is largelyconcerned with symplectic embeddings X , → α X ◦ where X is a concave or con-vex toric domain and α >
1. The definitions imply that concave or convex toricdomains X always satisfy X ⊂ α X ◦ for all α > X into α X ◦ . However we will findthat in many cases there are other such embeddings that are inequivalent to theinclusion in the following sense: Definition . Let A ⊂ U ⊂ C n , with A closed and U open, and let φ : A → U bea symplectic embedding. We say that φ is unknotted if there is a symplecto-morphism Ψ : U → U such that Ψ ( A ) = φ ( A ) . We say that φ is knotted if it isnot unknotted. Since A may not be a manifold or even a manifold with boundary we should say what itmeans for φ : A → U to be a symplectic embedding; our convention will be that it means thatthere is an open neighborhood of A to which φ extends as a symplectic embedding. When A is a manifold with boundary it is not hard to see using a relative Moser argument that this isequivalent to the statement that φ : A → U is a smooth embedding of manifolds with boundarywhich preserves the symplectic form. YMPLECTICALLY KNOTTED CODIMENSION-ZERO EMBEDDINGS OF DOMAINS IN R F IGURE
1. The shaded regions are examples of choices of Ω suchthat Theorem 1.8 gives knotted embeddings X Ω → α X ◦ Ω for suit-able α >
1. The dashed lines delimit the regions which are as-sumed to contain ( ∂ Ω ) ∩ ( ∞ ) in, respectively, Cases (i) and(ii) of the theorem.Note that we do not require the map Ψ to be compactly supported, or Hamil-tonian isotopic to the identity, or even to extend continuously to the closure of U ; accordingly our definition of knottedness is in principle more restrictive thanothers that one might use.In Section 1.1 (based on results from Sections 2 and 3) we will prove theexistence of knotted embeddings from X to α X ◦ for many toric domains X ⊂ C and suitable α > Theorem 1.8.
Let X ⊂ C belong to any of the following classes of domains:(i) All convex toric domains X such that, for some c > B ( c ) ( X ⊂ P ( c , c ) .(ii) All concave toric domains X Ω such that, for some c > { ( x , y ) ∈ [ ∞ ) | min { x + y , x + y } ≤ c } ⊂ Ω ( { ( x , y ) ∈ [ ∞ ) | x + y ≤ c } .(iii) All complex ℓ p balls { ( w , z ) ∈ C || w | p + | z | p ≤ r p } for p > log 9log 6 ≈ p = P ( a , b ) for a ≤ b < a .Then there exist α > φ : X → α X ◦ .For context, recall that McDuff showed in [ M91 ] that the space of symplecticembeddings from one four-dimensional ball to another is always connected; bythe symplectic isotopy extension theorem this implies that symplectic embed-dings B ( c ) → α B ( c ) ◦ can never be knotted. (In particular the exclusion of B ( c ) from each of the classes (i),(ii),(iii) above is necessary.) McDuff’s resultwas later extended to establish the connectedness of the space of embeddingsof one four-dimensional ellipsoid into another [ M09 ] or of a four-dimensional JEAN GUTT AND MICHAEL USHER concave toric domain into a convex toric domain [ C14 ] . So Theorem 1.8 reflectsthat embeddings from concave toric domains into concave ones, or convex toricdomains into convex ones, can behave differently than embeddings from con-cave toric domains into convex ones.We do not know whether the bound b < a in part (iv) of Theorem 1.8 issharp. The bound p > log 9log 6 in part (iii) is not sharp; we are aware of extensionsof our methods that lower this bound slightly, though in the interest of brevitywe do not include them. Note that the domains in part (iii) are concave when p < p > R , we show in The-orem 2.21 that the embeddings from Cases (i) and (iv) of Theorem 1.8 remainknotted after being trivially extended to the product of X Ω with an ellipsoid ofsufficiently large Gromov width. It remains an interesting problem to find knot-ted embeddings involving broader classes of high-dimensional domains that donot arise from lower-dimensional constructions.By the way, embeddings such as those in Theorem 1.8 can only be knotted fora limited range of α , since the extension-after-restriction principle [ S, Proposi-tion A.1 ] implies that for any compact set X ⊂ C n which is star-shaped withrespect to the origin and contains the origin in its interior and any symplecticembedding φ : X → C n , there is α > φ ( X ) ⊂ α X ◦ and such that φ is unknotted when considered as a map to α X ◦ for all α ≥ α . The values for α that we find in the proof of Theorem 1.8 vary from case to case, but in eachinstance lie between 1 and 2. This suggests the: Question 1.9.
Do there exist a domain X ⊂ R n , a number α >
2, and a knottedsymplectic embedding φ : X → α X ◦ ?Theorem 1.8 concerns embeddings of a domain X into the interior of a dilate α X ◦ of X ; of course it is also natural to consider embeddings in which the sourceand target are not simply related by a dilation. Our methods in principle allowfor this, though the proofs that the embeddings are knotted become more subtle.In Section 4 we carry this out for embeddings of four-dimensional polydisks intoother polydisks, and in particular we prove the following as Corollary 4.7: Theorem 1.10.
Given any y ≥
1, there exist polydisks P ( a , b ) and P ( c , d ) andknotted embeddings of P ( a , b ) into P ( y ) ◦ and of P ( y ) into P ( c , d ) ◦ .Theorem 1.10 and Case (iv) of Theorem 1.8 should be compared to [ FHW94,Section 3.3 ] , in which it is shown that, if a ≤ b < c but a + b > c , thenthe embeddings φ , φ : P ( a , b ) → P ( c , c ) ◦ given by φ ( w , z ) = ( w , z ) and φ ( w , z ) = ( z , w ) are not isotopic through compactly supported symplectomor-phisms of P ( c , c ) ◦ . However our embeddings are different from these; in factthe embeddings from [ FHW94 ] are not even knotted in our (rather strong)sense since there is a symplectomorphism of the open polydisk P ( c , c ) ◦ map-ping P ( a , b ) to P ( b , a ) . If one instead considers embeddings into P ( c , d ) with c < d chosen such that P ( c , d ) ◦ contains both P ( a , b ) and P ( b , a ) and a + b > d , YMPLECTICALLY KNOTTED CODIMENSION-ZERO EMBEDDINGS OF DOMAINS IN R then P ( a , b ) and P ( b , a ) are inequivalent to each other under the symplecto-morphism group of P ( c , d ) ◦ . However in situations where this construction andthe construction underlying Theorem 1.8 (iv) and Theorem 1.10 both apply,our knotted embeddings represent different knot types than both P ( a , b ) and P ( b , a ) , see Remark 4.5.Let us be a bit more specific about how we prove Theorem 1.8; the proof ofTheorem 1.10 is conceptually similar. The knotted embeddings φ : X → α X ◦ described in Theorem 1.8 are obtained as compositions of embeddings X → E → α X ◦ where E is an ellipsoid. In the cases that X is convex, the first map X → E is just an inclusion, while the second map E → α X ◦ is obtained by us-ing recent developments from [ M09 ] , [ C14 ] that ultimately have their roots inTaubes-Seiberg-Witten theory, see Section 3. (For a limited class of convex toricdomains X that are close to a cube P ( c , c ) , we provide a much more elementaryand explicit construction in Section 3.2.) In the cases that X is concave the re-verse is true: E → α X ◦ is an inclusion while X → E is obtained from these morerecent methods. Meanwhile, we use the properties of transfer maps in filtered S -equivariant symplectic homology to obtain a lower bound on possible values α such that there can exist any unknotted embedding X → α X ◦ which factorsthrough an ellipsoid E . In each case in Theorem 1.8, we will find compositions X → E → α X ◦ arising from the constructions in Section 3 for which α is lessthan this symplectic-homology-derived lower bound, leading to the conclusionthat the composition must be knotted. Figure 2 and its caption explain thismore concretely in a representative special case.To carry this out systematically, let us introduce the following two quanti-ties associated to a star-shaped domain X ⊂ C n , where the symbol , → alwaysdenotes a symplectic embedding:(1.2) δ ell ( X ) = inf { α ≥ | ( ∃ a , . . . , a n )( X , → E ( a , . . . , a n ) , → α X ◦ ) } and(1.3) δ uell ( X ) = inf (cid:26) α ≥ (cid:12)(cid:12)(cid:12)(cid:12) ( ∃ a , . . . , a n , f : X , → E ( a , . . . , a n ) , g : E ( a , . . . , a n ) , → α X ◦ )( g ◦ f is unknotted. ) (cid:27) (The u in δ uell stands for “unknotted.”) To put this into a different context, as wassuggested to us by Y. Ostrover and L. Polterovich, one can define a pseudometricon the space of star-shaped domains in C n by declaring the distance betweentwo domains X and Y to be the logarithm of the infimal α ∈ R such that there isa sequence of symplectic embeddings α − / X , → Y , → α / X ◦ ; a more refinedversion of this pseudometric would additionally ask that neither of the resultingcompositions X → α X ◦ and Y → α Y ◦ be knotted. Then (at least if n =
2) thelogarithm of δ ell ( X ) or of δ uell ( X ) is the distance from X to the set of ellipsoidswith respect to such a pseudometric. (In the case of δ uell this statement dependspartly on the result from [ M09 ] that when E is an ellipsoid in R a symplecticembedding E , → α E ◦ is never knotted.)We will prove Theorem 1.8 by proving, for each X as in the statement, astrict inequality δ ell ( X ) < δ uell ( X ) . This entails finding upper bounds for δ ell ( X ) JEAN GUTT AND MICHAEL USHER F IGURE
2. The strategy underlying our knotted embedding inthe case that X is the ℓ ball of capacity 1, as in Case (i) or (iii)of Theorem 1.8. X is the toric domain associated to the smallerregion on the left; the toric domain associated to the triangle onthe left is the ellipsoid E = E (( / ) / , 3 / ) , which in particularcontains X . The larger region at right is obtained by dilating X by α = ( + ǫ )( / ) / for a small ǫ >
0, and Proposition3.5 shows that there is a symplectic embedding φ : E → α X ◦ (in fact, φ has image contained in the preimage under µ of theinscribed quadrilateral on the right). Our knotted embedding is φ | X ; Theorem 1.12(a) implies that any unknotted embedding X → α X ◦ that extends to a symplectic embedding E → α X ◦ would have α ≥ / , whereas in this construction α can betaken arbitrarily close to ( / ) / .by exhibiting particular compositions of embeddings X , → E , → α X ◦ , and find-ing lower bounds for δ uell ( X ) using filtered positive S -equivariant symplectichomology. As it happens, for convex or concave toric domains both our upperbounds and our lower bounds can be conveniently expressed in terms of thefollowing notation: Notation 1.11.
For a domain Ω ⊂ [ ∞ ) n we define functions k · k ∗ Ω and [ · ] Ω from R n to R as follows: • For ~α ∈ R n , k ~α k ∗ Ω = sup { ~α · ~ v | ~ v ∈ Ω } . • For ~α ∈ R n , [ ~α ] Ω = inf { ~α · ~ v | ~ v ∈ [ ∞ ) n \ Ω } .The estimates for δ uell that are relevant to Theorem 1.8 are given by the fol-lowing result, proven in Section 2: YMPLECTICALLY KNOTTED CODIMENSION-ZERO EMBEDDINGS OF DOMAINS IN R Theorem 1.12. (a) If X Ω ⊂ C is a convex toric domain, then δ uell ( X Ω ) ≥ k (
1, 1 ) k ∗ Ω max {k (
1, 0 ) k ∗ Ω , k (
0, 1 ) k ∗ Ω } .(b) If X Ω ⊂ C is a concave toric domain, then δ uell ( X Ω ) ≥ min { [(
2, 1 )] Ω , [(
1, 2 )] Ω } [(
1, 1 )] Ω .As for upper bounds on δ ell , in Section 3.1 we prove the following: Theorem 1.13. (a) Suppose that Ω ⊂ [ ∞ ) is a domain such that ˆ Ω is convexand such that Ω contains points ( a , 0 ) , ( b ) , ( x , y ) with 0 < x ≤ a ≤ b ≤ x + y .Then δ ell ( X Ω ) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:129) a , 1 x + y ‹(cid:13)(cid:13)(cid:13)(cid:13) ∗ Ω .(b) Suppose that Ω ⊂ [ ∞ ) is a domain that contains (
0, 0 ) in its interiorand whose complement in [ ∞ ) is convex, and such that points ( a , 0 ) , ( b ) , ( x , y ) with 0 < x + y ≤ a ≤ b all belong to [ ∞ ) \ Ω . Then δ ell ( X Ω ) ≤ ”€ b , x + y Š— Ω .(c) For a polydisk P ( a , b ) with a ≤ b ≤ a we have δ ell ( P ( a , b )) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:129) a + b , 12 a + b ‹(cid:13)(cid:13)(cid:13)(cid:13) ∗ [ a ] × [ b ] .Assuming Theorems 1.12 and 1.13 for the time being, we now show howthey lead to Theorem 1.8.1.1. Proof of Theorem 1.8.
In each of the four cases it suffices to prove a strictinequality δ ell ( X ) < δ uell ( X ) .First let X = X Ω be a convex toric domain with B ( c ) ( X Ω ⊂ P ( c , c ) . Thus Ω ⊂ [ c ] × [ c ] (since X Ω ⊂ P ( c , c ) ), and Ω is a convex region containingthe points ( c , 0 ) , ( c ) , and (due to the strict inclusion B ( c ) ( X Ω ) some point ( x , y ) having x + y > c . The fact that ( c , 0 ) , ( c ) ∈ Ω ⊂ [ c ] × [ c ] impliesthat k (
1, 0 ) k ∗ Ω = k (
0, 1 ) k ∗ Ω = c . Consequently by Theorem 1.12(a), δ uell ( X Ω ) ≥ c k (
1, 1 ) k ∗ Ω .Meanwhile Theorem 1.13(a) gives δ ell ( X Ω ) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:129) c , 1 x + y ‹(cid:13)(cid:13)(cid:13)(cid:13) ∗ Ω = c (cid:13)(cid:13)(cid:13)(cid:13)(cid:129) cx + y ‹(cid:13)(cid:13)(cid:13)(cid:13) ∗ Ω .So to prove that δ ell ( X Ω ) < δ uell ( X Ω ) it suffices to show that k (
1, 1 ) k ∗ Ω > k ( a ) k ∗ Ω where a : = cx + y <
1. Choose ( v , v ) ∈ Ω such that v + av = ( a ) · ( v , v ) = k ( a ) k ∗ Ω ; it suffices to find ( w , w ) ∈ Ω with (
1, 1 ) · ( w , w ) = w + w > v + av . Bearing in mind that ( v , v ) ∈ Ω ⊂ [ ∞ ) and a <
1, if v = JEAN GUTT AND MICHAEL USHER we can simply take ( w , w ) = ( v , v ) . On the other hand if v = Ω ⊂ [ c ] × [ c ] we have v + av ≤ c , so taking ( w , w ) = ( x , y ) gives w + w > c ≥ v + av . So in any case k (
1, 1 ) k ∗ Ω ≥ w + w > v + av = k ( a ) k ∗ Ω ,proving that δ uell ( X Ω ) > δ ell ( X Ω ) and thus completing the proof of Case (i) ofTheorem 1.8.Case (ii) is rather similar. The hypothesis implies that all points ( x , y ) of [ ∞ ) \ Ω have min { x + y , x + y } ≥ c and so Theorem 1.12(b) yields δ uell ( X Ω ) ≥ c [( )] Ω . The hypothesis also implies that [ ∞ ) \ Ω contains a point ( x , y ) with x + y < c , and also contains the points ( c , 0 ) and ( c ) , so byTheorem 1.13(b) we have δ ell ( X Ω ) ≤ ”€ c , x + y Š— Ω = c ”€ cx + y Š— Ω .So to show that δ ell ( X Ω ) < δ uell ( X Ω ) it suffices to show that [( b )] Ω > [(
1, 1 )] Ω where b : = cx + y >
1. This is established in basically the same way as thesimilar inequality in Case (i): let ( v , v ) ∈ [ ∞ ) \ Ω minimize ~ v ( b ) · ~ v .Then either v =
0, in which case (
1, 1 ) · ( v , v ) < ( b ) · ( v , v ) , or else v = v = c by our assumptions on Ω , and so [(
1, 1 )] Ω ≤ x + y < c =( b ) · ( v , v ) . So in Case (ii) we indeed have δ ell ( X Ω ) < δ uell ( X Ω ) .We now turn to Case (iii) concerning complex ℓ p balls X = { ( w , z ) ∈ C | | w | p + | z | p ≤ r p } . Using appropriate rescalings it suffices to prove the result in the casethat r = p π , so that X = X Ω where Ω = { ( x , y ) ∈ [ ∞ ) | x p / + y p / ≤ } .When p > X Ω is a convex toric domain contained in P (
1, 1 ) and strictlycontaining B ( ) , so the result follows from Case (i). From now on assumethat 0 < p <
2, so that X Ω is a concave toric domain. Since p / <
1, thereverse Hölder inequality (and the fact that it is sharp) implies that for any ( v , w ) ∈ [ ∞ ) we have [( v , w )] Ω = ( v q + w q ) / q where q = pp − <
0. So fromTheorem 1.12(b) we obtain δ uell ( X ) ≥ ( q + ) / q / q = (cid:129) q − + ‹ − / | q | .Meanwhile [ ∞ ) \ Ω contains the points (
0, 1 ) , (
1, 0 ) , ( − / p , 2 − / p ) , so The-orem 1.13(b) yields δ ell ( X ) ≤ (cid:0) + (cid:0) − / p (cid:1) q (cid:1) / q = (cid:129) + ‹ − / q = (cid:129) ‹ − / | q | .So we will have δ ell ( X ) < δ uell ( X ) provided that 2 q − + < , where q = pp − .This condition is equivalent to 2 q < , i.e. , p − p > log 3log 2 , i.e. , p > log 9log 6 .Turning finally to Case (iv), let X = P ( a , b ) = X Ω where Ω = [ a ] × [ b ] andwe assume that a ≤ b < a . Clearly for ( v , w ) ∈ [ ∞ ) we have k ( v , w ) k ∗ Ω = av + bw . Hence Theorem 1.12(a) gives δ uell ( P ( a , b )) ≥ a + bb YMPLECTICALLY KNOTTED CODIMENSION-ZERO EMBEDDINGS OF DOMAINS IN R while Theorem 1.13(c) gives δ ell ( P ( a , b )) ≤ aa + b + b a + b .In other words, writing s = ba , we have δ uell ( P ( a , b )) ≥ + s and δ ell ( P ( a , b )) ≤ + + s + s + s . So δ ell ( P ( a , b )) < δ uell ( P ( a , b )) provided that 4 s + s < + s + s , i.e. provided that ba = s <
2, as we have assumed. (cid:3)
Organization of the paper.
The following Section 2 will recall some factsabout S -equivariant symplectic homology and extend these using an inverselimit construction to open subsets of R n in order to prove Theorem 1.12, whichis the key to showing that our embeddings are indeed knotted. The point of theargument, roughly speaking, is that the filtered positive S -equivariant sym-plectic homology of an ellipsoid E is “as simple as possible” given the total (un-filtered) homology, while that of the domains in Theorem 1.8 has additionalfeatures in the form of elements that persist over certain finite action intervalsbefore disappearing in the total homology. The ratios of the endpoints of theseintervals are related to the bounds that we prove on the quantity δ uell in Theo-rem 1.12. We also show that our knotted embeddings remain knotted in certainproducts in Section 2.1.The embeddings appearing in our main results are constructed in Section 3using methods derived from Taubes-Seiberg-Witten theory in work of McDuff [ M09 ] and Cristofaro-Gardiner [ C14 ] . While these sophisticated methods seemto be necessary to obtain results as broad as Theorems 1.8 and 1.10, we showin Section 3.2 that for certain domains that are close to a cube the embeddingscan be obtained by much more elementary methods, leading to explicit formulaswhich we provide. Section 4 extends the work in the rest of the paper to obtainthe knotted polydisks from Theorem 1.10.The appendix contains a proof of a lemma concerning filtered positive S -equivariant symplectic homology, showing that it can be identified as the filteredhomology of a certain filtered complex generated by good Reeb orbits. Thislemma probably will not surprise experts (in particular it was anticipated in [ GH17, Remark 3.2 ] ), and is similar to [ GG16, Proposition 3.3 ] , but we havenot seen full details of a proof of a result as sharp as this one elsewhere.1.3. Acknowledgements.
This work grew out of our consideration of a ques-tion of Yaron Ostrover and Leonid Polterovich. We are grateful to Richard Hind,Mark McLean, Yaron Ostrover, Leonid Polterovich, and Felix Schlenk for veryuseful discussions at various stages of this project. The work was partially sup-ported by the NSF through grant DMS-1509213 and by an AMS-Simons travelgrant.
2. O
BSTRUCTIONS TO UNKNOTTEDNESS FROM FILTERED POSITIVE S - EQUIVARIANT SYMPLECTIC HOMOLOGY
The goal of this section is to prove Theorem 1.12, which gives lower boundson the quantity δ uell defined in (1.3). The main tool for proving this theoremis the positive S -equivariant symplectic homology which was introduced byViterbo [ Vit99 ] and developed by Bourgeois and Oancea [ BO16, BO13a, BO13b,BO10 ] . We refer to [ BO16, BO13a, G15, GH17 ] for a precise definition, butdescribe here some of the key features.Let ( X , λ ) be a Liouville domain, so that X is a compact smooth manifoldwith boundary and λ ∈ Ω ( X ) has the properties that d λ is non-degenerateand that λ | ∂ X is a contact form. We say that ( X , λ ) is non-degenerate if thelinearized return map of the Reeb flow at each closed Reeb orbit on ∂ X , actingon the contact hyperplane ker λ , does not have 1 as an eigenvalue. We will alsoassume that the first Chern class of T X vanishes on π ( X ) .In this situation, as in [ GH17 ] , for each L ∈ R we have an L -filtered positive S -equivariant symplectic homology, denoted by C H L ( X , λ ) ; these are Q -vectorspaces that come equipped with maps ı L , L : C H L ( X , λ ) → C H L ( X , λ ) for L ≤ L such that ı L , L is the identity and ı L , L ◦ ı L , L = ı L , L . The assumptionon c ( T X ) implies that the C H L ( X , λ ) are Z -graded. The (unfiltered) positive S -equivariant symplectic homology of ( X , λ ) is C H ( X , λ ) = lim −→ L C H L ( X , λ ) where the direct limit is constructed using the maps ı L , L .The analysis of the spaces C H L ( X , λ ) is significantly simplified by the follow-ing, which is proven in the appendix. A slightly weaker version for a differentversion of S -equivariant symplectic homology is given in [ GG16, Proposition3.3 ] . Lemma 2.1.
Assume as above that ( X , λ ) is a non-degenerate Liouville domainwith c ( T X ) | π ( X ) =
0. There is an R -filtered chain complex (cid:0) C C ∗ ( X , λ ) , ∂ (cid:1) ,freely generated over Q by the good Reeb orbits of λ | ∂ X with the genera-tor corresponding to a Reeb orbit γ having filtration level equal to the ac-tion R γ λ and grading equal to the Conley-Zehnder index of γ , such that foreach k ∈ Z and L ∈ R the space C H Lk ( X , λ ) is isomorphic to the k th homol-ogy of the subcomplex C C L ∗ ( X , λ ) of C C ∗ ( X , λ ) consisting of elements withfiltration level at most L , and such that for L ≤ L the image of the map ı L , L : C H L k ( X , λ ) → C H L k ( X , λ ) is isomorphic to the image of the inclusion-induced map H k (cid:0) C C L ∗ ( X , λ ) (cid:1) → H k (cid:0) C C L ∗ ( X , λ ) (cid:1) .Moreover, the boundary operator ∂ on C C ∗ ( X , λ ) strictly decreases filtra-tion, in the sense that if x ∈ C C L ∗ ( X , λ ) then there is ǫ > ∂ x ∈ C C L − ǫ ∗ ( X , λ ) . Warning: In [ GH17 ] the map that we denote by ı L , L is denoted by ı L , L . Recall that a Reeb orbit γ is bad if it is an even degree multiple cover of another Reeb orbit γ ′ such that the Conley-Zehnder indices of γ and γ ′ have opposite parity. Otherwise, γ is good. YMPLECTICALLY KNOTTED CODIMENSION-ZERO EMBEDDINGS OF DOMAINS IN R Definition . A tame domain in R n is a 2 n -dimensional Liouville domain ( X , λ ) where: • X is a compact submanifold with boundary of R n ; • d λ = ω , where ω = P ni = d x i ∧ d y i is the (restriction of the) standardsymplectic form on R n ; and • the Reeb flow of λ | ∂ X is non-degenerate.A tame star-shaped domain is a subset X ⊂ R n such that ( X , λ | X ) is a tamedomain, where λ = X i ( x i d y i − y i d x i ) .Said differently, a tame star-shaped domain is a smooth star-shaped domainsuch that the radial vector field on R n is transverse to the boundary, and suchthat the characteristic flow on the boundary is non-degenerate. Remark . If U ⊂ R n is open and λ ∈ Ω ( U ) with d λ = ω , and if X ⊂ U hasthe property that ( X , λ | X ) is a tame domain, we will typically write C H L ( X , λ ) instead of C H L ( X , λ | X ) . It should be noted however that C H L ( X , λ ) dependsonly on the restriction of λ to X . In fact, more specifically, given that we alwaysassume that d λ = ω the only dependence of C H L ( X , λ ) on λ (as opposed to d λ ) arises from the germ of λ | X along ∂ X ; this feature is part of what allowsfor the construction of transfer maps associated to generalized Liouville embed-dings in [ GH17 ] .Let ( X , λ ) and ( X ′ , λ ′ ) be two non-degenerate Liouville domains. If φ : X , → ( X ′ ) ◦ is a symplectic embedding with the property that ( φ ⋆ λ ′ − λ ) | ∂ X is exact ,then for all L ∈ R , there exists a map Φ L φ : C H L ( X ′ , λ ′ ) −→ C H L ( X , λ ) called the transfer map. This map is defined in [ GH17, Section 8.1 ] . If X ⊂ ( X ′ ) ◦ , we will simply write Φ L for the transfer map induced by the inclusion of X into X ′ .Such a transfer map Φ L φ also exists in the case that, instead of being a gen-eralized Liouville embedding into the interior of X ′ , φ : X → X ′ is simply anisomorphism of Liouville domains ( i.e. φ is a diffeomorphism with φ ⋆ λ ′ = λ ).In this case Φ L φ is an isomorphism.The transfer map is functorial in the sense that if ( X , λ ) , ( X , λ ) , and ( X , λ ) are tame domains and if φ : X , → X and ψ : X , → X are eithergeneralized Liouville embeddings or isomorphisms of Liouville domains, thenthe following diagram is commutative:(2.1) C H L ( X , λ ) Φ L ψ / / Φ L ψ ◦ φ C H L ( X , λ ) Φ L φ / / C H L ( X , λ ) . Such embeddings in general are called “generalized Liouville embeddings” of X into X ′ . (This is proven in the unfiltered context for Liouville embeddings in [ G15, The-orem 4.12 ] , and the same argument proves the result in our more general situ-ation.)Recall that a tame star-shaped domain W by definition has the property that ( W , λ ) is a non-degenerate Liouville domain, where λ is the standard Liouvilleprimitive P i ( x i d y i − y i d x i ) , so in this case we obtain graded vector spaces C H L ( W , λ ) . In this case, for any ζ >
0, the scaled domain ζ W = { p ζ ~ x | ~ x ∈ W } is likewise a tame domain with respect to λ . By pulling back the ingre-dients in the construction of C H L ( W , λ ) by appropriate rescalings, we obtainan identification of C H L ( W , λ ) with C H ζ L ( ζ W , λ ) (on the level of the Reeborbits that generate the complex C C ∗ ( W , λ ) , this sends an orbit γ : S → R n to the orbit p ζγ , which has the effect of multiplying the action by ζ ). We callthis isomorphism C H L ( W , λ ) ∼ = C H ζ L ( ζ W , λ ) the “rescaling isomorphism.”The following gives useful relations between this rescaling isomorphism andthe other maps in the theory. Lemma 2.4.
Let W be a tame star-shaped domain, ζ >
1, and 0 < s < t . Thenthe diagrams(2.2) C H ζ − t ( W , λ ) ı ζ − t , t / / rescaling ∼ = (cid:15) (cid:15) C H t ( W , λ ) C H t ( ζ W , λ ) Φ t ❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ and(2.3) C H s ( W , λ ) ı s , t / / C H t ( W , λ ) rescaling ∼ = (cid:15) (cid:15) C H ζ s ( ζ W , λ ) ı ζ s , ζ t / / rescaling ∼ = O O C H ζ t ( ζ W , λ ) are both commutative. Proof.
The commutativity of (2.3) follows by conjugating the various ingredi-ents involved in the construction of
C H by rescalings, see [ G15, Lemma 4.15 ] .The commutativity of (2.2) follows from the description of the transfer mor-phism Φ t : C H t ( ζ W , λ ) → C H t ( W , λ ) in [ G15, Lemma 4.16 ] ; indeed it isshown there that the chain map which induces Φ t on filtered homology can bechosen to be the one that sends an orbit near the boundary of ζ W to its imageunder the rescaling ζ W → W , and this correspondence multiplies actions by ζ − . (cid:3) YMPLECTICALLY KNOTTED CODIMENSION-ZERO EMBEDDINGS OF DOMAINS IN R Lemma 2.5.
Let X be a tame star-shaped domain. Let b ≥ a >
0. Then thefollowing diagram is commutative:
C H L ( bX , λ ) Φ L / / ≃ rescaling (cid:15) (cid:15) C H L ( aX , λ ) ≃ rescaling (cid:15) (cid:15) C H b − L ( X , λ ) ı b − L , a − L / / C H a − L ( X , λ ) . Proof.
Consider the diagram
C H L ( bX , λ ) Φ L / / C H L ( aX , λ ) ∼ = (cid:15) (cid:15) C H ab L ( aX , λ ) ∼ = h h ◗◗◗◗◗◗◗◗◗◗◗◗ ı ab L , L ♠♠♠♠♠♠♠♠♠♠♠♠ C H b − L ( X , λ ) ∼ = O O ∼ = ♥♥♥♥♥♥♥♥♥♥♥♥ ı b − L , a − L / / C H a − L ( X , λ ) where all of the indicated isomorphisms are given by rescaling. The left trian-gle commutes trivially, the upper triangle commutes as a special case of (2.2),and the lower right quadrilateral commutes as a special case of (2.3). Hencethe entire diagram commutes, which implies the result since the left map is anisomorphism. (cid:3) Lemma 2.6.
Let X , X ′ ⊂ R n and λ ∈ Ω ( X ′ ) be such that X ⊂ ( X ′ ) ◦ and both ( X , λ | X ) and ( X ′ , λ | X ′ ) are tame domains, and let Ψ be a symplectomorphismbetween open subsets of R n whose domain contains X ′ . Then the followingdiagram is commutative:(2.4) C H L ( X ′ , λ ) Φ L / / ≃ Φ L Ψ − (cid:15) (cid:15) C H L ( X , λ ) ≃ Φ L Ψ − (cid:15) (cid:15) C H L (cid:0) Ψ ( X ′ ) , Ψ − ⋆ λ (cid:1) Φ L / / C H L (cid:0) Ψ ( X ) , Ψ − ⋆ λ (cid:1) Proof.
This is a direct consequence of the functoriality (2.1): writing i : X → X ′ and j : Ψ ( X ) → Ψ ( X ′ ) for the inclusions, we have a commutative diagram Ψ ( X ) j / / Ψ − (cid:15) (cid:15) Ψ ( X ′ ) Ψ − (cid:15) (cid:15) X i / / X ′ and (2.4) is obtained by taking transfer maps. (cid:3) In proving Theorem 1.12 it will be helpful to know that the image of the map ı L , L is not too small in certain situations. The following two lemmas give ourfirst results in this direction. Lemma 2.7.
Let X Ω be a convex toric domain in C . Then for any δ , ǫ > X δ , ǫ Ω such that ( − ǫ ) X Ω ⊂ X δ , ǫ Ω ⊂ X ◦ Ω andsuch that, for any L , L withmax (cid:8) k (
1, 0 ) k ∗ Ω , k (
0, 1 ) k ∗ Ω (cid:9) + δ ≤ L < L ≤ k (
1, 1 ) k ∗ Ω − δ ,the map ı L , L : C H L ( X δ , ǫ Ω , λ ) −→ C H L ( X δ , ǫ Ω , λ ) is an isomorphism of two-dimensional vector spaces. Proof.
The constructions in steps 1, 2, and 3 of [ GH17, Proof of Lemma 2.5 ] use a Morse-Bott perturbation of a suitable smoothing of X Ω to obtain a tamestar-shaped domain X δ , ǫ Ω that can be arranged to have the properties that ( − ǫ ) X Ω ⊂ X δ , ǫ Ω ⊂ X ◦ Ω and such that the Reeb orbits of λ | ∂ X δ , ǫ Ω having action atmost k (
1, 1 ) k ∗ Ω and Conley-Zehnder index at most 4 consist of: • no orbits of index 2; • two orbits of index 3, with actions in the intervals (cid:0) k (
1, 0 ) k ∗ Ω − δ , k (
1, 0 ) k ∗ Ω + δ (cid:1) and (cid:0) k (
0, 1 ) k ∗ Ω − δ , k (
0, 1 ) k ∗ Ω + δ (cid:1) , respectively; and • at most one orbit of index 4, with action greater than k (
1, 1 ) k ∗ Ω − δ .So letting C C L ∗ ( X δ , ǫ Ω , λ ) be as in Lemma 2.1 (so that in particular C H Lk ( X δ , ǫ Ω , λ ) ∼ = H k (cid:0) C C L ∗ ( X δ , ǫ Ω , λ ) , ∂ (cid:1) ), for any L in (cid:2) max {k (
1, 0 ) k ∗ Ω , k (
0, 1 ) k ∗ Ω } + δ , k (
1, 1 ) k ∗ Ω − δ (cid:3) we have C C L ( X δ , ǫ Ω , λ ) = C C L ( X δ , ǫ Ω , λ ) = { } and C C L ( X δ , ǫ Ω , λ ) ∼ = Q , andmoreover if L , L both lie in this interval with L < L then the inclusion ofcomplexes C C L ( X δ , ǫ Ω , λ ) → C C L ( X δ , ǫ Ω , λ ) is an isomorphism. So passing tohomology shows that, for max {k (
1, 0 ) k ∗ Ω , k (
0, 1 ) k ∗ Ω } + δ ≤ L < L ≤ k (
1, 1 ) k ∗ Ω − δ , the inclusion-induced map ı L , L : C H L ( X δ , ǫ Ω , λ ) → C H L ( X δ , ǫ Ω , λ ) is an iso-morphism of two-dimensional vector spaces. (cid:3) Lemma 2.8.
Let X Ω be a concave toric domain in C . Then for any δ , ǫ > X δ , ǫ Ω such that ( − ǫ ) X Ω ⊂ X δ , ǫ Ω ⊂ X ◦ Ω andsuch that, if [(
1, 1 )] Ω + δ ≤ L < L ≤ min { [(
1, 2 )] Ω , [(
2, 1 )] Ω } − δ ,the map ι L , L : C H L ( X δ , ǫ Ω , λ ) −→ C H L ( X δ , ǫ Ω , λ ) is an isomorphism of one-dimensional vector spaces. Proof.
We argue analogously to the proof of Lemma 2.7. By [ GH17, Proof ofLemma 2.7 ] , there is a tame star-shaped domain X δ , ǫ Ω such that ( − ǫ ) X Ω ⊂ X δ , ǫ Ω ⊂ X ◦ Ω and such that the part of C C ∗ ( X δ , ǫ Ω , λ ) of filtration level at mostmax { [(
1, 2 )] Ω , [(
2, 1 )] Ω } and degree at most five is generated by: YMPLECTICALLY KNOTTED CODIMENSION-ZERO EMBEDDINGS OF DOMAINS IN R • one generator, denoted a , in degree 3, with filtration level in theinterval ([(
1, 1 )] Ω − δ , [(
1, 1 )] Ω + δ ) ; • one generator, denoted b , in degree 4, with filtration level in theinterval ([(
1, 1 )] Ω − δ , [(
1, 1 )] Ω + δ ) ; and • at most two generators c and c in degree 5, with respective filtra-tion levels in the intervals (cid:0) [(
1, 2 )] Ω − δ , [(
1, 2 )] Ω + δ (cid:1) and (cid:0) [(
2, 1 )] Ω − δ , [(
2, 1 )] Ω + δ (cid:1) .Moreover it is a standard fact (see e.g. [ GH17, Proposition 3.1 ] ) that the fulldegree-3 homology C H ( X δ , ǫ Ω , λ ) of this complex is isomorphic to Q ; indeedthis statement holds for arbitrary tame star-shaped domains in R . Also, [ GH17,Theorem 1.14 ] shows that a generator for C H ( X δ , ǫ Ω , λ ) is represented by achain having filtration level at most [(
1, 1 )] Ω . So since the generator a spansthe part of C C ( X δ , ǫ Ω , λ ) with filtration level at most max { [(
1, 2 )] Ω , [(
2, 1 )] Ω } (which is greater than [(
1, 1 )] Ω ), it follows that a must not be in the image ofthe boundary operator ∂ . Since the boundary operator preserves the filtration,we must then have ∂ b = [(
1, 1 )] Ω + δ ≤ L ≤ min { [(
1, 2 )] Ω , [(
2, 1 )] Ω } − δ , the element b isa degree-four cycle in the subcomplex C C L ∗ ( X δ , ǫ Ω , λ ) , which is not a boundaryfor the trivial reason that, for this range of L , C C L ( X δ , ǫ Ω , λ ) = { } . Thus b de-scends to homology to generate the one-dimensional vector space C H L ( X δ , ǫ Ω , λ ) for any such L , and the map ı L , L : C H L ( X δ , ǫ Ω , λ ) → C H L ( X δ , ǫ Ω , λ ) is an iso-morphism whenever [(
1, 1 )] Ω + δ ≤ L < L ≤ min { [(
1, 2 )] Ω , [(
2, 1 )] Ω }− δ . (cid:3) We are now going to extend the definition of
C H L to open subsets of R n .This is part of what makes it possible to prove knottedness in the strong senseof Defnition 1.7, which considers arbitrary symplectomorphisms of the open setthat serves as the codomain for the embedding. Working with open sets alsoallows us to consider domains with poorly-behaved boundaries, to which thestandard definition of C H L does not apply.We continue to denote by ω the standard symplectic form P ni = d x i ∧ d y i on open subsets of R n . Definition . Let U be an open subset of R n and let λ ∈ Ω ( U ) be such that d λ = ω . We define the positive S -equivariant symplectic homology of ( U , λ ) as(2.5) C H L ( U , λ ) : = lim ←− ( X , λ | X ) tame domain X ⊂ U C H L ( X , λ ) .Here the inverse limit is taken over transfer maps Φ L associated to inclusions.Given open sets U ⊂ V ⊂ R n and λ ∈ Ω ( V ) with d λ = ω , we define atransfer map Φ L : C H L ( V , λ ) → C H L ( U , λ ) as the inverse limit of transfer maps Φ L : C H L ( Y , λ ) → C H L ( X , λ ) as X , Y vary through sets such that ( X , λ ) , ( Y , λ ) are both tame with X ⊂ U ∩ Y ◦ and Y ⊂ V . This construction will be extended to certain other symplectic embeddings of open subsets (not just inclusions) inLemma 2.18. Lemma 2.10. If X is a tame star-shaped domain and if L is not the action of anyperiodic Reeb orbit on ∂ X then the natural map C H L ( X , λ ) → C H L ( X ◦ , λ ) isan isomorphism. Proof.
The system of tame star-shaped domains { ( − ǫ ) X | ǫ > } is cofinal inthe system of all tame star-shaped domains Y with Y ⊂ X ◦ , so there is a naturalisomorphism C H L ( X ◦ , λ ) ∼ = lim ←− ǫ> C H L (cid:0) ( − ǫ ) X , λ (cid:1) .Lemma 2.5 then induces a natural isomorphismlim ←− ǫ> C H L (cid:0) ( − ǫ ) X , λ (cid:1) ∼ = lim ←− ǫ> C H ( − ǫ ) − L ( X , λ ) where the inverse limit on the right is constructed from the maps ı s , t that areidentified by Lemma 2.1 with the maps induced by inclusions of subcomplexes C C s ∗ , → C C t ∗ . Since L is not the action of any periodic Reeb orbit on ∂ X , it fol-lows from Lemma 2.1 that the map ı L , ( − ǫ ) − L : C H L ( X , λ ) → C H ( − ǫ ) − L ( X , λ ) is an isomorphism for all sufficiently small ǫ , from which the lemma immedi-ately follows. (cid:3) Let U be an open subset of C n and λ ∈ Ω ( U ) with d λ = ω , and let L < L ∈ R . We define the map ı L , L : C H L ( U , λ ) → C H L ( U , λ ) as the inverselimit of the maps ı L , L : C H L ( X U , λ | X U ) → C H L ( X U , λ | X U ) where ( X U , λ | X U ) isa tame domain, X U ⊂ U .Since the inverse limit is a functor from the category of diagrams of abeliangroups to the category of abelian groups (see [ Wei94, Application 2.6.7 ] ), wehave a similar statement to Lemma 2.4: Lemma 2.11.
Let U be an open set in R n , let ζ >
1, and let λ ∈ Ω ( U ) with d λ = ω . Then the following diagram is commutative:(2.6) C H ζ − L ( U , λ ) ı ζ − L , L / / rescaling (cid:15) (cid:15) C H L ( U , λ ) C H L ( ζ U , λ ) Φ L ❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦ .The following calculation related to the maps ı L , L will be very helpful. Lemma 2.12.
Let X Ω be a convex toric domain in C .(i) If max {k (
1, 0 ) k ∗ Ω , k (
0, 1 ) k ∗ Ω } < L < k (
1, 1 ) k ∗ Ω , then C H L ( X ◦ Ω , λ ) is atwo-dimensional vector space.(ii) If max {k (
1, 0 ) k ∗ Ω , k (
0, 1 ) k ∗ Ω } < L < L < k (
1, 1 ) k ∗ Ω , then ı L , L : C H L ( X ◦ Ω , λ ) → C H L ( X ◦ Ω , λ ) is an isomorphism. YMPLECTICALLY KNOTTED CODIMENSION-ZERO EMBEDDINGS OF DOMAINS IN R Proof.
Choose δ > {k (
1, 0 ) k ∗ Ω , k (
0, 1 ) k ∗ Ω } + δ < L , L , L < k (
1, 1 ) k ∗ Ω − δ .For this fixed value of δ and varying ǫ >
0, the non-degenerate domains X δ , ǫ Ω from Lemma 2.7 form a cofinal system in the inverse system defining C H L ( X ◦ ω , λ ) .Choose a sequence ǫ m ց X δ , ǫ m Ω ⊂ ( X δ , ǫ m + Ω ) ◦ for each m , so we havetransfer maps ( Φ L ) m : C H L ( X δ , ǫ m + Ω , λ ) → C H L ( X δ , ǫ m Ω , λ ) ; this gives a cofinalsubsystem within our inverse system. We claim that these transfer maps are iso-morphisms of two-dimensional vector spaces once m is sufficiently large (andhence ǫ m is sufficiently small).To prove this, we first note that the domain and codomain both have dimen-sion two by Lemma 2.7, so it is enough to show that ( Φ L ) m is injective for alllarge m . But we have inclusions ( − ǫ m ) X δ , ǫ m + Ω ⊂ ( − ǫ m ) X ◦ Ω ⊂ ( X δ , ǫ m Ω ) ◦ ⊂ X δ , ǫ m Ω ⊂ ( X δ , ǫ m + Ω ) ◦ and so the transfer map ( Φ L ) m fits into a sequence of transfer maps(2.7) C H L ( X δ , ǫ m + Ω , λ ) ( Φ L ) m / / C H L ( X δ , ǫ m Ω , λ ) / / C H L (cid:0) ( − ǫ m ) X δ , ǫ m + Ω , λ (cid:1) whose composition is identified up to isomorphism by Lemma 2.4 with theinclusion-induced map ı L , ( − ǫ m ) − L : C H L ( X δ , ǫ m + Ω , λ ) → C H ( − ǫ m ) − L ( X δ , ǫ m + Ω , λ ) .Provided that m is chosen so large that ( − ǫ m ) − L < k (
1, 1 ) k ∗ Ω − δ , Lemma 2.7shows that the above map ı L , ( − ǫ ) − L is an isomorphism. Thus for m sufficientlylarge the first map ( Φ L ) m in the sequence (2.7) must be injective, and hence isalso an isomorphism by counting dimensions.Since the ( Φ L ) m are all isomorphisms for m sufficiently large, and since theyform the structure maps in a cofinal system within the inverse system defining C H L ( X ◦ Ω , λ ) , it follows that the canonical map C H L ( X ◦ Ω , λ ) → C H L ( X δ , ǫ m Ω , λ ) is an isomorphism for m sufficiently large. So by Lemma 2.7 C H L ( X ◦ Ω , λ ) istwo-dimensional, proving statement (i) of the lemma. Moreover this argumentworks uniformly for all L in the interval from max {k (
1, 0 ) k ∗ Ω , k (
0, 1 ) k ∗ Ω } + δ to k (
1, 1 ) k ∗ Ω − δ , and in particular for L = L or L = L where L , L are as instatement (ii) of the lemma. So for sufficiently large m we have a commutativediagram C H L ( X ◦ Ω , λ ) ı L L / / (cid:15) (cid:15) C H L ( X ◦ Ω , λ ) (cid:15) (cid:15) C H L ( X δ , ǫ m Ω , λ ) ı L L / / ı L L / / C H L ( X δ , ǫ m Ω , λ ) where the vertical arrows are isomorphisms by what we have just shown, andthe bottom horizontal arrow is an isomorphism by Lemma 2.7. Hence the tophorizontal arrow is an isomorphism, proving statement (ii) of the lemma. (cid:3) Lemma 2.13.
Let X Ω be a concave toric domain in C such that [(
1, 1 )] Ω < max { [(
1, 2 )] Ω , [(
2, 1 )] Ω } . Then for [(
1, 1 )] Ω < L < L < min { [(
1, 2 )] Ω , [(
2, 1 )] Ω } , ı L , L : C H L ( X ◦ Ω , λ ) −→ C H L ( X ◦ Ω , λ ) is an isomorphism of one-dimensional vector spaces. Proof.
This follows by the exact same argument as Lemma 2.12, using Lemma2.8 instead of Lemma 2.7. (cid:3)
Remark . In the case that X Ω is an ellipsoid E ( a , b ) (and hence in particularis both a concave toric domain and a convex toric domain), Lemmas 2.12 and2.13 have no content when applied to X Ω . Indeed in this case, assuming withoutloss of generality that a ≤ b , k (
1, 1 ) k ∗ Ω = max {k (
1, 0 ) k ∗ Ω , k (
0, 1 ) k ∗ Ω } = b , [(
1, 1 )] Ω = min { [(
1, 2 )] Ω , [(
2, 1 )] Ω = a and so there are no choices of L , L that satisfy the hypotheses. For each ofthe domains appearing in our main theorem, on the other hand, Lemma 2.12or Lemma 2.13 gives important information. Proposition 2.15.
Let U ⊂ R n be a star-shaped open set, and let φ : U → V be a symplectomorphism where V is an open subset of R n . Then φ determinesan isomorphism Φ L φ : C H L ( V , φ − ⋆ λ ) → C H L ( U , λ ) such that the diagram(2.8) C H L ( V , φ − ⋆ λ ) Φ L / / Φ L φ (cid:15) (cid:15) C H L (cid:0) φ ( W ) , φ − ⋆ λ (cid:1) Φ L φ | W (cid:15) (cid:15) C H L ( U , λ ) Φ L / / C H L ( W , λ ) commutes when W ⊂ U is an open subset. Proof.
For X ⊂ U , it is straightforward to see that ( X , λ ) is a non-degenerateLiouville domain if and only if (cid:0) φ ( X ) , φ − ⋆ λ (cid:1) is a non-degenerate Liouvilledomain. So in view of Lemma 2.6, we obtain an isomorphism of the inversesystems defining C H L ( V , φ − ⋆ λ ) and C H L ( U , λ ) . This induces the desiredisomorphism Φ L φ between the inverse limits C H L ( V , φ − ⋆ λ ) and C H L ( U , λ ) ,and the fact that (2.8) commutes follows by taking inverse limits of the diagrams(2.4) from Lemma 2.6. (cid:3) Definition . Let U ⊂ R n be an open subset and let λ ∈ Ω ( U ) obey d λ = ω .We say that the pair ( U , λ ) is tamely exhausted if for every compact subset K ⊂ U there is a set X with K ⊂ X ⊂ U such that ( X , λ ) is a tame Liouvilledomain and such that the natural map H ( X ; R ) → H ( ∂ X ; R ) is zero. YMPLECTICALLY KNOTTED CODIMENSION-ZERO EMBEDDINGS OF DOMAINS IN R Example 2.17.
In any dimension m , let us say that a nonempty compact subset X ⊂ R m is strictly star-shaped if for all x ∈ X and all t ∈ [
0, 1 ) it holds that t x ∈ X ◦ . We claim that if X ⊂ R n is strictly star-shaped then ( X ◦ , λ ) is tamelyexhausted.To see this, first note that for any a ∈ S n − the set I a = { t ≥ | ta ∈ X } is a closed interval of the form [ f ( a )] where 0 < f ( a ) < ∞ . Indeed I a contains all sufficiently small positive numbers because the definition impliesthat 0 ∈ X ◦ , and I a is closed and bounded because X is compact. So we cantake f ( a ) = sup I a = max I a ; the fact that I a contains all numbers between 0and f ( a ) is an obvious consequence of the assumption that X is star-shaped.Moreover we then have t f ( a ) a ∈ X ◦ for all t ∈ [
0, 1 ) .So we have defined a function f : S n − → ( ∞ ) with the properties that X = (cid:8) sa | a ∈ S n − , 0 ≤ s ≤ f ( a ) (cid:9) and X ◦ = (cid:8) sa | a ∈ S n − , 0 ≤ s < f ( a ) (cid:9) .We will now show that f is continuous. Let a ∈ S n − and let ǫ > f ( a ) > ǫ . Then (cid:0) f ( a ) − ǫ (cid:1) a ∈ X ◦ , so by considering a small ballaround (cid:0) f ( a ) − ǫ (cid:1) a that is contained in X ◦ we see that, for b ∈ S n − sufficientlyclose to a , it will hold that (cid:0) f ( a ) − ǫ (cid:1) b ∈ X ◦ and hence that f ( b ) > f ( a ) − ǫ .Thus f is lower semi-continuous. To see that f is upper semi-continuous notethat if it were not then we could find a k , a ∈ S n − with a k → a and each f ( a k ) ≥ f ( a ) + ǫ for some ǫ > k . Since X is compact, afterpassing to a subsequence the f ( a k ) a k would converge to a point of the form sa where both s > f ( a ) and sa ∈ X , contradicting the defining property of f . So f is indeed continuous.With this in hand it is not hard to see that our strictly star-shaped domain X istamely exhausted. Indeed, if K is a compact subset of X ◦ then there is ǫ > a ∈ S n − and t ≥ ta ∈ K , we have t < f ( a ) − ǫ . Choose a C ∞ function g : S n − → ( ∞ ) such that, for all a ∈ S n − , f ( a ) − ǫ < g ( a ) < f ( a ) .Then defining Y = (cid:8) ta | a ∈ S n − , 0 ≤ t ≤ g ( a ) (cid:9) , Y will be a smooth manifoldwith boundary such that λ | ∂ Y is a contact form and such that K ⊂ Y ⊂ X ◦ .Possibly after a further perturbation of g , the Reeb flow of λ | ∂ Y will be non-degenerate so that ( Y , λ ) is tame. Because Y is star-shaped, it obviously has H ( Y ; R ) =
0. Since K is an arbitrary compact subset of X ◦ this proves ourclaim that ( X ◦ , λ ) is tamely exhausted. Lemma 2.18.
To each symplectic embedding φ : U , → V between open subsets U , V ⊂ R n equipped with one-forms λ , λ ′ such that ( U , λ ) and ( V , λ ′ ) are tamelyexhausted, we may associate a map Φ L φ : C H L ( V , λ ′ ) → C H L ( U , λ ) such that:(i) In the case that φ is the inclusion of U into V and λ = λ ′ | U , Φ L φ co-incides with the transfer map Φ L : C H L ( V , λ ′ ) → C H L ( U , λ ′ ) describedjust before Lemma 2.10. (ii) If ( U , λ ) , ( V , λ ′ ) , ( W , λ ′′ ) are tamely exhausted and if φ : U , → V and ψ : V , → W are symplectic embeddings then we have a commutativediagram(2.9) C H L ( W , λ ′′ ) Φ L ψ ◦ φ / / Φ L ψ ' ' ❖❖❖❖❖❖❖❖❖❖❖ C H L ( U , λ ) C H L ( V , λ ′ ) Φ L φ ♣♣♣♣♣♣♣♣♣♣♣ Proof.
Since ( U , λ ) is tamely exhausted, the subsets X ⊂ U with ( X , λ ) tameand H ( X ; R ) → H ( ∂ X ; R ) zero form a cofinal system in the inverse systemdefining C H L ( U , λ ) . So in order to construct Φ L φ it suffices to define maps Φ LV X : C H L ( V , λ ′ ) → C H L ( X , λ ) for all such X in such a way that the diagrams(2.10) C H L ( V , λ ′ ) Φ LVX (cid:15) (cid:15) Φ LVX ′ ' ' ❖❖❖❖❖❖❖❖❖❖❖ C H L ( X , λ ) Φ L / / C H L ( X ′ , λ ) commute for subsets X , X ′ ⊂ U as above with X ′ ⊂ X ◦ .To define Φ LV X , note that the fact that ( V , λ ′ ) is tamely exhausted implies thatthere is Y with φ ( X ) ⊂ Y ◦ ⊂ Y ⊂ V such that ( Y , λ ′ ) is tame, and define Φ V X as a composition
C H L ( V , λ ′ ) → C H L ( Y , λ ′ ) → C H L ( X , λ ) where the first mapis the structure map of the inverse limit and the second map is the transfermap associated to φ | X : X , → Y ◦ . (The fact that φ | X is a generalized Liouvilleembedding follows from the facts that φ preserves ω and that H ( X ; R ) → H ( ∂ X ; R ) vanishes.)We claim that this map Φ LV X is independent of the choice of Y involved inits construction. Indeed if Y ′ ⊂ V is another set satisfying the same properties,then the fact that ( V , λ ′ ) is tamely exhausted shows that there is Z such that Y ∪ Y ′ ⊂ Z ◦ ⊂ Z ⊂ V and such that ( Z , λ ′ ) is tame. We can then form acommutative diagram C H L ( V , λ ′ ) ' ' ❖❖❖❖❖❖❖❖❖❖❖ + + ❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲ (cid:31) (cid:31) ❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄ C H L ( Z , λ ′ ) (cid:15) (cid:15) / / C H L ( Y , λ ′ ) (cid:15) (cid:15) C H L ( Y ′ , λ ′ ) / / C H L ( X , λ ) .Every piece of the above diagram (the square and the two triangles) is commu-tative by definition of the inverse limit and by functoriality of the transfer map.Therefore the two compositions C H L ( V , λ ′ ) → C H L ( X , λ ) passing respectivelythrough C H L ( Y , λ ′ ) and C H L ( Y ′ , λ ′ ) are equal to each other. YMPLECTICALLY KNOTTED CODIMENSION-ZERO EMBEDDINGS OF DOMAINS IN R To see that (2.10) commutes, notice that, by what we have just shown, wemay use the same subdomain Y ⊂ V in the constructions of Φ LV X and of Φ LV X ′ ,yielding a commutative diagram C H L ( V , λ ′ ) Φ LVX (cid:0) (cid:0) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) (cid:15) (cid:15) Φ LVX ′ (cid:31) (cid:31) ❃❃❃❃❃❃❃❃❃❃❃❃❃❃❃❃❃❃❃ C H L ( Y , λ ′ ) w w ♣♣♣♣♣♣♣♣♣♣♣ ' ' ❖❖❖❖❖❖❖❖❖❖❖ C H L ( X , λ ) Φ L / / C H L ( X ′ , λ ) where the bottom triangle is an instance of (2.1). So passing to the inverse limitover X indeed yields our desired map Φ L φ : C H L ( V , λ ′ ) → C H L ( U , λ ) .It remains to show that the various maps Φ L φ construced in this way satisfyproperties (i) and (ii) in the statement of the lemma. However, given the valid-ity of the above construction of Φ L φ and the functoriality (2.1) for transfer mapsassociated to generalized Liouville embeddings, both of these are straightfor-ward exercises with inverse limits and so we leave them to the reader. (cid:3) Corollary 2.19.
Let X , E , Y ⊂ R n be strictly star-shaped domains with X ⊂ Y ◦ ,and let f : X → E , g : E → Y ◦ be symplectic embeddings. If the composition g ◦ f is unknotted, then for all k ∈ Z , L ∈ R it holds thatRank (cid:0) Φ L : C H Lk ( Y ◦ , λ ) → C H Lk ( X ◦ , λ ) (cid:1) ≤ dim C H Lk ( E ◦ , λ ) . Proof.
The assumption that g ◦ f is unknotted implies that there is a symplec-tomorphism φ : Y ◦ → Y ◦ such that φ ( g ( f ( X ))) = X . Example 2.17 showsthat each of ( X ◦ , λ ) , ( E ◦ , λ ) , and ( Y ◦ , λ ) is tamely exhausted. It is clear fromthe definition that if ψ : Z , → R n is a symplectic embedding with image Z ′ ,then ( Z , ψ ⋆ λ ) is tamely exhausted if and only if ( Z ′ , λ ) is tamely exhausted. Sosince our symplectomorphism φ maps Y ◦ to Y ◦ and g (cid:0) f ( X ◦ ) (cid:1) to X ◦ it followsthat ( Y ◦ , φ ⋆ λ ) and (cid:0) g ( f ( X ◦ )) , φ ⋆ λ (cid:1) are also tamely exhausted. Consider thediagram C H Lk ( E ◦ , λ ) Φ L ( g | f ( X ◦ )) − ) ) ❘❘❘❘❘❘❘❘❘❘❘❘❘ C H Lk ( Y ◦ , φ ⋆ λ ) Φ L / / Φ Lg ♥♥♥♥♥♥♥♥♥♥♥♥ Φ L φ − (cid:15) (cid:15) C H Lk (cid:0) g ( f ( X ◦ )) , φ ⋆ λ (cid:1) Φ L φ − (cid:15) (cid:15) C H Lk ( Y ◦ , λ ) Φ L / / C H Lk ( X ◦ , λ ) We see that the top triangle commutes since it is an instance of (2.9) (as g ◦ ( g | f ( X ◦ ) ) − is just the inclusion of g ( f ( X ◦ )) into Y ◦ ); the square commutes by Corollary 2.15; and the vertical arrows are isomorphisms. Hence for each k ∈ Z ,Rank (cid:0) Φ L : C H Lk ( Y ◦ , λ ) → C H Lk ( X ◦ , λ ) (cid:1) = Rank (cid:0) Φ L : C H Lk ( Y ◦ , φ ⋆ λ ) → C H Lk ( g ( f ( X ◦ )) , φ ⋆ λ ) (cid:1) .But since Φ L : C H Lk ( Y ◦ , φ ⋆ λ ) → C H Lk ( g ( f ( X ◦ )) , φ ⋆ λ ) factors through C H Lk ( E ◦ , λ ) ,its rank is at most the dimension of C H Lk ( E ◦ , λ ) . (cid:3) Throughout the rest of the paper Corollary 2.19 will be our main tool forshowing that embeddings are knotted. First we need the following to showthat it applies to the domains appearing in our main theorems.
Proposition 2.20.
Let X be either a convex toric domain or a concave toricdomain in R n . Then X is strictly star-shaped. Proof.
First suppose that X = X Ω is a convex toric domain; thus Ω ⊂ [ ∞ ) n has the property that ˆ Ω (as defined in (1.1)) is a convex domain in R n . It is easyto see that X Ω is strictly star-shaped if and only if ˆ Ω is strictly star-shaped.Let us re-emphasize that “domains” are by definition closures of boundedopen sets. Consequently if x = ( x , . . . , x n ) ∈ ˆ Ω and 0 < ǫ <
1, we can find ( y , . . . , y n ) ∈ ˆ Ω ◦ such that y i x i > − ǫ for all i such that x i =
0. Now ˆ Ω ◦ is convexand is invariant under reversal of the sign of any subset of the coordinates of R n , so it follows that (cid:8) ( z , . . . , z n ) | | z i | ≤ y i (cid:9) ⊂ ˆ Ω ◦ . In particular this impliesthat ( − ǫ ) x ∈ ˆ Ω ◦ . Since ǫ can be taken arbitrarily small this proves that ˆ Ω isstrictly star-shaped and hence that X Ω is strictly star-shaped.Now let us turn to the case that X = X Ω is a concave toric domain, so that Ω ⊂ [ ∞ ) n has the property that [ ∞ ) n \ Ω is convex. It is easy to see that X Ω is strictly star-shaped if and only if Ω has the property that t Ω ⊂ Ω ◦ forall t ∈ [
0, 1 ) , where the interior Ω ◦ is taken relative to [ ∞ ) n . Suppose forcontradiction that x = ( x , . . . , x n ) ∈ Ω and t x / ∈ Ω ◦ where 0 ≤ t <
1. Then t x ∈ [ ∞ ) n \ Ω ◦ = [ ∞ ) n \ Ω .Now [ ∞ ) n \ Ω is a convex set which (since Ω is compact) contains all pointssufficiently far from the origin in addition to containing t x , in view of which (cid:8) ( y , . . . , y n ) | y i ≥ t x i for all i (cid:9) ⊂ [ ∞ ) n \ Ω .The set on the left hand side above contains our point x in its interior, so wewould have x ∈ Ω ∩ € [ ∞ ) n \ Ω Š ◦ .But € [ ∞ ) n \ Ω Š ◦ = ([ ∞ ) n \ Ω ◦ ) ◦ = [ ∞ ) n \ Ω ◦ ,so we would have x ∈ Ω ∩ (cid:0) [ ∞ ) n \ Ω ◦ (cid:1) , which is impossible since Ω is theclosure of an open subset. (cid:3) We now fulfill the main goal of this section by proving Theorem 1.12.
YMPLECTICALLY KNOTTED CODIMENSION-ZERO EMBEDDINGS OF DOMAINS IN R Proof of Theorem 1.12 (a).
We will show that α > δ uell ( X ) implies that α ≥ k ( ) k ∗ Ω max {k ( ) k ∗ Ω , k ( ) k ∗ Ω . Let α > δ uell ( X ) . Then there is an ellipsoid E and embed-dings f : X , → E and g : E , → α X ◦ such that g ◦ f is unknotted. By slightlyperturbing E we may assume that E is irrational ( i.e. E = E ( a , b ) where ba / ∈ Q ); this ensures that E is a tame star-shaped domain. We will apply Corol-lary 2.19 with k =
3. Note that, for each L ∈ R , dim C H L ( E ◦ , λ ) ≤ [ BCE07, Section 3 ] . So by Corollary 2.19, we must haveRank ( Φ L : C H L ( α X ◦ , λ ) → C H L ( X ◦ , λ )) ≤ L ∈ R . By Lemma 2.11,then, ı α − L , L : C H α − L ( X ◦ , λ ) → C H L ( X ◦ , λ ) has rank at most one.If we had α < k ( ) k ∗ Ω max {k ( ) k ∗ Ω , k ( ) k ∗ Ω } , then Lemma 2.12 would allow us to finda real number L such that ı α − L , L : C H α − L ( X ◦ , λ ) → C H L ( X ◦ , λ ) is an iso-morphism of two-dimensional vector spaces, a contradiction which proves that α ≥ k ( ) k ∗ Ω max {k ( ) k ∗ Ω , k ( ) k ∗ Ω } , as desired. (cid:3) Proof of Theorem 1.12 (b).
This follows by essentially the same argument, using k = k =
3, and appealing toLemma 2.13 instead of Lemma 2.12. This yields the result since any irrationalellipsoid E has no periodic Reeb orbits on its boundary with Conley-Zehnderindex equal to 4 and hence obeys C H L ( E ◦ , λ ) = { } for all L ∈ R . (cid:3) Products.
The goal of this section is to show that Theorem 1.8 extends toproducts of convex toric domains with large ellipsoids of arbitrary even dimen-sion.
Theorem 2.21.
Let X ⊂ C belong to any of the following classes of domains:(i) All convex toric domains X such that, for some c > B ( c ) ( X ⊂ P ( c , c ) .(ii) All polydisks P ( a , b ) for a ≤ b < a .Then there exist numbers α > R > φ : X × E ( b , . . . , b n − ) → α (cid:0) X × E ( b , . . . , b n − ) (cid:1) ◦ for any b , . . . , b n − with each b i ≥ R .(Specific values for R in the various cases will appear in the proof.)In order to prove this we will first establish some basic facts concerningthe relationship of the filtered positive S -equivariant symplectic homology ofa product of two convex toric domains to that of the factors. Observe thatthe product of two convex toric domains is a convex toric domain: we have X Ω × X Ω = X Ω × Ω . Also notice that, if Ω ⊂ R m and Ω ⊂ R n and if weexpress general elements of R n + m as ( α , β ) where α ∈ R m and β ∈ R n , then k ( α , β ) k ∗ Ω × Ω = k α k ∗ Ω + k β k ∗ Ω . Proposition . Let X Ω ⊂ C and X Ω ⊂ C n − be two convex toric domains,and assume that min {k e i k ∗ Ω } > k (
1, 1 ) k ∗ Ω where { e , . . . , e n − } is the standardbasis for R n − . Then for any δ , ǫ > Z δ , ǫ such that ( − ǫ ) X Ω × Ω ⊂ Z δ , ǫ ⊂ X ◦ Ω × Ω and such that, formax ¦ k (
1, 0 ) k ∗ Ω , k (
0, 1 ) k ∗ Ω © + δ ≤ L < L ≤ k (
1, 1 ) k ∗ Ω − δ ,the map ı L , L : C H L n + ( Z δ , ǫ , λ ) −→ C H L n + ( Z δ , ǫ , λ ) is an isomorphism of two-dimensional vector spaces. Proof.
As in the proof of Lemma 2.7, Steps 1, 2, and 3 of the proof of [ GH17,Lemma 2.5 ] provide a tame star-shaped domain Z δ , ǫ such that ( − ǫ ) X Ω × Ω ⊂ Z δ , ǫ ⊂ X ◦ Ω × Ω and such that the Reeb orbits of λ | ∂ Z δ , ǫ having action at most k (
1, 1 ) k ∗ Ω and Conley-Zehnder index at most n + • no orbits of index n ; • two orbits (
1, 0 ) , (
0, 1 ) in degree n +
1, with actions in the intervals € k (
1, 0 ) k ∗ Ω − δ , k (
1, 0 ) k ∗ Ω + δ Š and € k (
0, 1 ) k ∗ Ω − δ , k (
0, 1 ) k ∗ Ω + δ Š , re-spectively; and • at most one orbit (
1, 1 ) of index n +
2, with filtration level greater than k (
1, 1 ) k ∗ Ω − δ .(In general we would potentially obtain orbits with actions approximately k ( α , β ) k ∗ Ω × Ω = k α k ∗ Ω + k β k ∗ Ω for arbitrary α ∈ N and β ∈ N n − , but our restriction to filtra-tion levels less than or equal to k (
1, 1 ) k ∗ Ω , which is assumed to be less than each k e i k ∗ Ω forces β to be zero.)So as in the proof of Lemma 2.7, for L in the interval (cid:2) max {k (
1, 0 ) k ∗ Ω , k (
0, 1 ) k ∗ Ω } + δ , k (
1, 1 ) k ∗ Ω − δ (cid:3) we have C C Ln ( Z δ , ǫ , λ ) = C C Ln + ( Z δ , ǫ , λ ) = { } and C C Ln + ( Z δ , ǫ , λ ) ∼ = Q , andmoreover if L , L both lie in this interval with L < L then the inclusion ofcomplexes C C L n + ( Z δ , ǫ , λ ) → C C L n + ( Z δ , ǫ , λ ) is an isomorphism. So passing tohomology shows that, for max {k (
1, 0 ) k ∗ Ω , k (
0, 1 ) k ∗ Ω } + δ ≤ L < L ≤ k (
1, 1 ) k ∗ Ω − δ , the inclusion-induced map ı L , L : C H L n + ( Z δ , ǫ , λ ) → C H L n + ( Z δ , ǫ , λ ) is anisomorphism of two-dimensional vector spaces. (cid:3) Lemma 2.23.
Let X Ω be a convex toric domain in C with the property thatmax ¦ k (
1, 0 ) k ∗ Ω , k (
0, 1 ) k ∗ Ω © < k (
1, 1 ) k ∗ Ω , and let X Ω be a convex toric domainin C n − such that min ≤ i ≤ n − k e i k ∗ Ω > k (
1, 1 ) k ∗ Ω . Then for all small η > (cid:18) C H max ¦ k ( ) k ∗ Ω , k ( ) k ∗ Ω © + η n + (cid:0) ( X Ω × X Ω ) ◦ , λ (cid:1) −→ C H k ( ) k ∗ Ω − η n + (cid:0) ( X Ω × X Ω ) ◦ , λ (cid:1)(cid:19) = Proof.
Given Proposition 2.22, this is proven in exactly the same way as Lemma2.12. (cid:3)
Lemma 2.24. If E ( a , a ) ⊂ R is an ellipsoid with 1 < a a / ∈ Q , and if E ( b , . . . , b n − ) ⊂ R n − is any ellipsoid, then dim C H Ln + €(cid:0) E ( a , a ) × E ( b , . . . , b n − ) (cid:1) ◦ , λ Š ≤ L < min { b i } . YMPLECTICALLY KNOTTED CODIMENSION-ZERO EMBEDDINGS OF DOMAINS IN R Proof.
Given p >
1, consider the Hamiltonian H : C × C n − → R : H ( z , w ) : = ‚(cid:18) π | z | a + π | z | a (cid:19) p + ‚ n − X i = π | w i | b i Œ p Œ p A computation shows the Hamiltonian vector field X H of H obeys λ ( X H ) = − H , from which one deduces that the Reeb vector field of λ along the boundaryof Z p : = { H ≤ } is equal to − X H . (Here we use the sign convention that defines X H by d λ ( X H , · ) = d H .)Note that Z p ⊂ E ( a , a ) × E ( b , . . . , b n − ) , and that (because the ℓ p norm on R converges uniformly on compact subsets to the ℓ ∞ norm as p → ∞ ), forany ǫ > ( − ǫ ) (cid:0) E ( a , a ) × E ( b , . . . , b n − ) (cid:1) ⊂ Z p for all sufficientlylarge p .The Reeb flow on ∂ Z p rotates the w i coordinates with period b i €P n − i = π | w i | b i Š − ( p − ) ,which is greater than or equal to b i since, on ∂ Z p , we have P n − i = π | w i | b i ≤ ∂ Z p having action less than min { b i } must haveall b i identically zero.Because a a / ∈ Q , it is easy to check that any closed Reeb orbit on ∂ Z p musthave one or both of z , z identically equal to zero. Such an orbit which also hasall b i equal to zero has action ka or ka where k ∈ N . Moreover the Conley-Zehnder index of such an orbit is given by 2 k + š ka a (cid:157) + n − k + š ka a (cid:157) + n −
1. Indeed the linearized flow splits into the symplectic sum of the linearizedflows on E ( a , a ) and on E ( b , . . . , b n − ) . Thus the Conley-Zehnder index isthe sum of the Conley-Zehnder indices of each individual linearized flow.In particular, there is only one such orbit of Conley-Zehnder index n + z plane and has all other coordi-nates equal to zero. It follows that Z p is arbitrarily well-approximated by non-degenerate star-shaped domains Z ǫ p ⊂ Z ◦ p such that, for L < min { b i } , we havedim C H Ln + ( Z ǫ p , λ ) ≤
1. By using these Z ǫ p for p ≫ (cid:0) E ( a , a ) × E ( b , . . . , b n − ) (cid:1) ◦ it is not hard to see (using arguments like the one in the proofof Lemma 2.12) that dim C H Ln + €(cid:0) E ( a , a ) × E ( b , . . . , b n − ) (cid:1) ◦ , λ Š ≤ (cid:3) Proof of Theorem 2.21.
In Case (a), the proof of Theorem 1.8 shows that δ ell ( X Ω ) < c k (
1, 1 ) k ∗ Ω . Hence there is a sequence of symplectic embeddings X Ω , → E ( a , a ) , → α X ◦ Ω where (without loss of generality) 1 < a a / ∈ Q and 1 < α < c k (
1, 1 ) k ∗ Ω = k ( ) k ∗ Ω max {k ( ) k ∗ Ω , k ( ) k ∗ Ω } . By taking a product with the identity, this yields symplecticembeddings X Ω × E ( b , . . . , b n − ) , → E ( a , a ) × E ( b , . . . , b n − ) , → α (cid:0) X Ω × E ( b , . . . , b n − ) (cid:1) ◦ .If the composition of these embeddings were unknotted, then Corollary 2.19(applied with L slightly smaller than k (
1, 1 ) k ∗ Ω ) and Lemma 2.23 would show that dim C H Ln + (( E ( a , a ) × E ( b , . . . , b n − ) ◦ , λ ) ≥
2, a contradiction with Lemma2.24 provided that we choose R ≥ k (
1, 1 ) k ∗ Ω .In Case (b), the proof of Theorem 1.8 likewise shows that there is a sequenceof symplectic embeddings P ( a , b ) , → E ( a , a ) , → α P ( a , b ) ◦ where 1 < a a / ∈ Q and 1 < α < a + bb = k ( ) k ∗ Ω max {k ( ) k ∗ Ω , k ( ) k ∗ Ω } . (Here we write Ω = [ a ] × [ b ] ).Then the same argument as in Case (a) applies to show that the product of thecomposition of these embeddings with the identity on E ( b , . . . , b n − ) will beknotted provided that b i ≥ R : = a + b for all i . (cid:3)
3. S
OME EMBEDDINGS OF FOUR - DIMENSIONAL ELLIPSOIDS
The main goal of this section is to prove Theorem 1.13, which asserts theexistence of certain symplectic embeddings to and from four-dimensional ellip-soids. The machinery for constructing (or, perhaps more accurately, ascertain-ing the existence of) such embeddings has its roots in Taubes-Seiberg-Wittentheory and in papers such as [ MP94 ] , [ M09 ] , [ C14 ] which relate the questionof whether certain four-dimensional domains symplectically embed into certainother domains to questions about symplectic ball-packing problems and then toquestions about the symplectic cones of blowups of C P , which are then con-verted to elementary problems by results from [ LiLi02 ] . We will presently recallsome of these results, rephrasing them in a way suitable for our applications.In this section we will consider a limited class of toric domains in C , givenas the preimage under the standard moment map µ : ( w , z ) ( π | w | , π | z | ) ofa quadrilateral having a right-angled vertex at the origin and satisfying a coupleof other conditions, see Figure 3. More specifically: Definition . Let a , b , x , y ∈ [ ∞ ) satisfy the following properties:(i) x ≤ a and y ≤ b .(ii) If xa + yb <
1, then x + y ≤ min { a , b } .(iii) If xa + yb >
1, then x + y ≥ max { a , b } .We denote by T ( a , b , x , y ) the preimage under µ of the quadrilateral in R having vertices (
0, 0 ) , ( a , 0 ) , ( x , y ) , ( b ) .Any such set T ( a , b , x , y ) is said to be a toric quadrilateral ; it is said to beconcave if xa + yb ≤ xa + yb ≥ a , b , x , y ∈ Q , then T ( a , b , x , y ) is said to be a rational toric quadrilateral .We allow the possibility that ( x , y ) lies on the line segment from ( a , 0 ) to ( b ) so that the relevant quadrilateral degenerates to a triangle; indeed in thiscase T ( a , b , x , y ) is the ellipsoid E ( a , b ) (and is both concave and convex).For any rational concave toric quadrilateral T ( a , b , x , y ) (and indeed for some-what more general toric domains), [ C14 ] (generalizing [ M09 ] ) explains howto construct the so-called “weight sequence” w (cid:0) T ( a , b , x , y ) (cid:1) of T ( a , b , x , y ) ,which is a finite unordered sequence of positive numbers. We will rephrase thisas follows. Given two unordered sequences of positive numbers a = [ a , . . . , a k ] , b =[ b , . . . , b l ] we write a ⊔ b for the union with repetitions: a ⊔ b = [ a , . . . , a k , b , . . . , b l ] . YMPLECTICALLY KNOTTED CODIMENSION-ZERO EMBEDDINGS OF DOMAINS IN R F IGURE
3. The images under µ of the toric quadrilaterals T (
5, 4, 4, 3 ) and T (
4, 5, 1, 2 ) .We will abbreviate the weight sequence w (cid:0) E ( a , b ) (cid:1) = w (cid:0) T ( a , b , a , 0 ) (cid:1) as W ( a , b ) .Then for a general rational concave toric quadrilateral the weight sequence isdetermined recursively by the following prescriptions: • For any a ≥ W ( a , 0 ) = W ( a ) = [] (the empty sequence). • For 0 < a ≤ b , W ( a , b ) = W ( b , a ) = [ a ] ⊔ W ( a , b − a ) . • If xa + yb < x + y ≤ min { a , b } ) then w (cid:0) T ( a , b , x , y ) (cid:1) = [ x + y ] ⊔ W ( a − x − y , y ) ⊔ W ( b − x − y , x ) .For instance, w ( T (
4, 5, 1, 2 )) = [ ] ⊔ W (
1, 2 ) ⊔ W (
2, 1 ) = [ ] ⊔ [ ] ⊔ W (
1, 1 ) ⊔ [ ] ⊔ W (
1, 1 )= [
3, 1, 1, 1, 1 ] ⊔ W (
1, 0 ) ⊔ W (
1, 0 ) = [
3, 1, 1, 1, 1 ] .Dually, to a convex toric quadrilateral T is associated a “weight expansion,”which takes the form of a pair (cid:0) h ( T ) ; ˆ w ( T ) (cid:1) where h ( T ) ∈ [ ∞ ) is called the“head” and ˆ w ( T ) is a possibly-empty unordered sequence of positive numbersand is called the “negative weight sequence.” For a general rational convex toricquadrilateral the weight expansion is determined as follows: • If a ≤ b then h (cid:0) E ( a , b ) (cid:1) = h (cid:0) E ( b , a ) (cid:1) = b and ˆ w (cid:0) E ( a , b ) (cid:1) = ˆ w (cid:0) E ( b , a ) (cid:1) = W ( b , b − a ) . • If xa + yb > x + y ≥ max { a , b } ), then h (cid:0) T ( a , b , x , y ) (cid:1) = x + y and ˆ w (cid:0) T ( a , b , x , y ) (cid:1) = W ( x + y − a , y ) ⊔ W ( x + y − b , x ) .(This is a complete prescription, since by definition any convex toric quadri-lateral T ( a , b , x , y ) has xa + yb ≥
1, with equality implying that T ( a , b , x , y ) = E ( a , b ) . A more obviously-consistent phrasing is that the head h ( T ) is equalto the capacity of the smallest ball containing T , and that the negative weightsequence is the union of the weight sequences of ellipsoids whose interiors areequivalent under the action of translations and S L ( Z ) to the components of B (cid:0) h ( T ) (cid:1) ◦ \ T .)The deep result that we need is: Theorem 3.2. [ C14, Theorem 1.4 ] Let S , . . . , S k be a rational concave toricquadrilaterals and T be a rational convex toric quadrilateral. Then the followingare equivalent: (i) For all α > ` ki = S i , → α T .(ii) For all α > k a i = a c ∈ w ( S i ) B ( c ) ! ⊔ a c ∈ ˆ w ( T ) B ( c ) , → B (cid:0) α h ( T ) (cid:1) .(While [ C14, Theorem 1.4 ] is stated for a single concave toric domain S , theproof—which closely follows the proof for ellipsoids in [ M09 ] —extends withoutchange to a collection of several disjoint such domains, as was already notedwhen all of the domains are ellipsoids in [ M11, Proposition 3.5 ] .)Let us introduce the following notation. If [ a , . . . , a m ] is an unordered se-quence of nonnegative real numbers, and if t is another nonnegative real num-ber, we will write [ a , . . . , a m ] (cid:22) [ t ] if and only if t ≥ inf ¨ u : m a i = B ( a i ) symplectically embeds into B ( u ) « Then Theorem 3.2 can be rephrased as stating that, for concave toric quadri-laterals S , . . . , S k and a convex toric quadrilateral T , the statement that forall α > ` ki = S i , → α T is equivalent to thestatement that w ( S ) ⊔ · · · ⊔ w ( S k ) ⊔ ˆ w ( T ) (cid:22) [ h ( T )] . Remark . As follows from [ MP94 ] and [ LiLiu01 ] , if we denote by H thehyperplane class and E , . . . , E m the exceptional divisors of the manifold X m obtained by blowing up C P m times, the statement that [ a , . . . , a m ] (cid:22) [ t ] is equivalent to the statement that the Poincaré dual of the class tH − P a i E i lies in the set ¯ C K given as the closure of the subset of H ( X m ; R ) consisting ofthe cohomology classes of symplectic forms having associated canonical classPoincaré dual to − H + P E i .We will often find it useful to combine Theorem 3.2 with the following ele-mentary but somewhat subtle fact. In the special case that b and c are integermultiples of a this has a well-known proof as in [ M09, Lemma 2.6 ] ; see also [ M11, Lemma 2.6 ] for a corresponding statement about ECH capacities in adifferent special case. Proposition 3.4.
Let a , b , c ∈ ( ∞ ) . Then for all α > E ( a , b ) ` E ( a , c ) , → α E ( a , b + c ) . Proof.
For any v , w > ƒ ( v , w ) = ( v ) × ( w ) and △ ( v , w ) = { ( x , x ) ∈ ( ∞ ) | x v + x w < } . Also for A , B ⊂ R write A × L B for the “La-grangian product” { ( x + i y , x + i y ) | ( x , x ) ∈ A , ( y , y ) ∈ B } ⊂ C . Now theTraynor trick [ T95, Corollary 5.3 ] shows that for all γ < γ E ( v , w ) , → △ ( v , w ) × L ƒ (
1, 1 ) . Conversely ( x + i y , x + i y ) YMPLECTICALLY KNOTTED CODIMENSION-ZERO EMBEDDINGS OF DOMAINS IN R ( q x π e π i y , q x π e π i y ) defines a symplectic embedding △ ( v , w ) × L ƒ (
1, 1 ) → E ( v , w ) . Meanwhile the symplectomorphism of C given by ( x + i y , x + i y ) ( v − x + iv y , w − x + iw y ) maps △ ( v , w ) × L ƒ (
1, 1 ) to △ (
1, 1 ) × L ƒ ( v , w ) . Hence:For any v , w > γ <
1, there are symplectic embeddings(3.1) γ E ( v , w ) , → △ (
1, 1 ) × L ƒ ( v , w ) , → E ( v , w ) .The proof readily follows from this: if γ <
1, we may symplectically embed E ( γ a , γ b ) , → △ (
1, 1 ) × L ƒ ( a , b ) = { ( x + i y , x + i y ) | x , x , y , y > x + x < y < a , y < b } ,and likewise, by composing an embedding as in (3.1) with a translation in the y direction, we may symplectically embed E ( γ a , γ c ) , → { ( x + i y , x + i y ) | x , x , y > x + x < y < a , b < y < b + c } .The images of these two embeddings are evidently disjoint, and their union iscontained in △ (
1, 1 ) × L ƒ ( a , b + c ) , which symplectically embeds into E ( a , b + c ) .We thus obtain, for any γ <
1, a symplectic embedding E ( γ a , γ b ) ` E ( γ a , γ c ) , → E ( a , b + c ) ; conjugation by a rescaling then gives the embeddings required inthe proposition. (cid:3) The following family of embeddings is used in Case (i) of Theorem 1.8; seeFigure 2 for more context in a particular instance.
Proposition 3.5.
Let a , b , x , y ∈ ( ∞ ) with x ≤ a , y ≤ b , and a ≤ b ≤ x + y .Then for all α > E ( a , x + y ) , → α T ( a , b , x , y ) . Proof.
It evidently suffices to prove the statement when a , b , x , y ∈ Q . Then byTheorem 3.2 the statement is equivalent to the statement that W ( a , x + y ) ⊔W ( x + y − a , y ) ⊔W ( x + y − b , x ) (cid:22) [ x + y ] . But another application of Theorem3.2 shows that this, in turn, is equivalent to the statement that for all α > E ( a , x + y ) ⊔ E ( x + y − a , y ) ⊔ E ( x + y − b , x ) , → α B ( x + y ) .Since we assume that a ≤ b (so x + y − b ≤ x + y − a ), we have symplecticembeddings E ( x + y − a , y ) ⊔ E ( x + y − b , x ) , → E ( x + y − a , y ) ⊔ E ( x + y − a , x ) , → p α E ( x + y − a , x + y ) where the first map is the inclusion and the second is given by Proposition 3.4.Combining this with another application of Proposition 3.4 yields: E ( a , x + y ) ⊔ E ( x + y − a , y ) ⊔ E ( x + y − b , x ) , → E ( a , x + y ) ⊔ p α E ( x + y − a , x + y ) ⊂ p α ( E ( a , x + y ) ⊔ E ( x + y − a , x + y )) , → α E ( x + y , x + y ) = α B ( x + y ) . (cid:3) Similarly in the concave case, we obtain:
Proposition 3.6.
Let a , b , x , y ∈ ( ∞ ) with x + y ≤ a ≤ b . Then for all α > T ( a , b , x , y ) , → E ( b , x + y ) . Proof.
It again suffices to assume that a , b , x , y ∈ Q . Theorem 3.2 shows thatthe proposition is equivalent to the statement that [ x + y ] ⊔ W ( a − x − y , y ) ⊔W ( b − x − y , x ) ⊔ W ( b − x − y , b ) (cid:22) [ b ] , which in turn is equivalent to theexistence of a symplectic embedding, for all α > B ( x + y ) ⊔ E ( a − x − y , y ) ⊔ E ( b − x − y , x ) ⊔ E ( b , b − x − y ) , → α B ( b ) .Proposition 3.4 (together with the inclusion E ( a − x − y , y ) ⊂ E ( b − x − y , y ) )gives, for all ν >
1, embeddings E ( a − x − y , y ) ⊔ E ( b − x − y , x ) , → ν E ( b − x − y , x + y ) and then B ( x + y ) ⊔ E ( b − x − y , x + y ) , → ν E ( b , x + y ) , andfinally E ( b , x + y ) ⊔ E ( b , b − x − y ) , → ν E ( b , b ) = ν B ( b ) . Combining thesethree embeddings (with ν = α / ) then implies the result. (cid:3) Remark . Note that the volume of T ( a , b , x , y ) is ( a y + b x ) , while that of E ( a , x + y ) is a ( x + y ) . So in the case that a = b , the embeddings E ( a , x + y ) → α T ( a , a , x , y ) (in the convex case) or T ( a , a , x , y ) → α E ( a , x + y ) (in theconcave case) fill all but an arbitrarily small proportion of the volumes of theirtargets as α → P ( b ) = T ( b , 1, b ) , a special case of Proposition 3.5 is that, for any α > b ≥
1, there is a symplectic embedding E (
1, 1 + b ) , → α P ( b ) . Thefollowing reproduces this embedding when 1 ≤ b <
2, and improves on it for b = m + ǫ ≥
2. The case that ǫ = [ CFS17, Remark 1.2(1) ] . Proposition 3.8.
Let m ∈ Z + and 0 ≤ ǫ <
1. Then for all α > E (
1, 2 m + ǫ ) , → α P ( m + ǫ ) . Proof.
By Theorem 3.2 the statement is equivalent to the statement that W (
1, 2 m + ǫ ) ⊔ W ( m + ǫ , m + ǫ ) ⊔ W (
1, 1 ) (cid:22) [ m + + ǫ ] . From the recursive descriptionof W ( a , b ) given earlier we see that W (
1, 2 m + ǫ ) = W ( m ) ⊔ W ( m + ǫ ) , sothis is equivalent to the existence, for all α >
1, of a symplectic embedding E ( m ) ⊔ E ( m + ǫ ) ⊔ E ( m + ǫ , m + ǫ ) ⊔ E (
1, 1 ) , → α B ( m + + ǫ ) .But by Proposition 3.4 there are symplectic embeddings E ( m ) ⊔ E (
1, 1 ) , → p α E ( m + ) ⊂ p α E ( m + + ǫ ) and E ( m + ǫ ) ⊔ E ( m + ǫ , m + ǫ ) , → p α E ( m + ǫ , m + + ǫ ) ,and then another application of Proposition 3.4 gives a symplectic embedding p α E ( m + ǫ , m + + ǫ ) ⊔p α E ( m + + ǫ ) , → α E ( m + + ǫ , m + + ǫ ) = α B ( m + + ǫ ) ,from which the result is immediate. (cid:3) The embeddings in Propositions 3.5, 3.6, and 3.8 will give rise to many of theknotted embeddings described in the introduction. Some of our other knottedembeddings require a somewhat less straightforward application of Theorem3.2 and Proposition 3.4. The key additional (and standard) ingredient is theuse of
Cremona moves , based on [ LiLi02, Proof of Lemma 3.4 ] . As in Remark3.3 we regard the question of whether [ a , . . . , a m ] (cid:22) [ t ] as equivalent to the YMPLECTICALLY KNOTTED CODIMENSION-ZERO EMBEDDINGS OF DOMAINS IN R question of whether the Poincaré dual of tH − P a i E i lies in the closure ¯ C K of the appropriate connected component of the symplectic cone of the m -foldblowup X m of C P . Since [ a , . . . , a m ] (cid:22) [ t ] if and only if [ a , . . . , a m , 0 ] (cid:22) [ t ] we may without loss of generality assume that m ≥
3. Then X m contains asphere in the class H − E − E − E of self-intersection − C K and maps the Poincaré dual of tH − P a i E i to the Poincaré dual of ( t − a − a − a ) H − ( t − a − a ) E − ( t − a − a ) E − ( t − a − a ) E − P mi = a i E i .So we have: Proposition 3.9. [ LiLi02 ] Assume that t ≥ max { a + a , a + a , a + a } . Then [ a , a , a , a , . . . , a m ] (cid:22) [ t ] if and only if [ t − a − a , t − a − a , t − a − a , a , . . . , a m ] (cid:22) [ t − a − a − a ] The following will help us construct the knotted polydisks from Case (iv) ofTheorem 1.8.
Proposition 3.10.
Let a ≤ y ≤ b ≤ a . Then for all α > E (cid:129) a + b a + y ‹ , → α T ( a , b , a , y ) . Proof.
As usual assuming that a , b , y ∈ Q , by Theorem 3.2 the proposition isequivalent to the statement that(3.2) W (cid:129) a + b a + y ‹ ⊔ W ( y , y ) ⊔ W ( a + y − b , a ) (cid:22) [ a + y ] .Since a ≤ y and b ≤ a we have a + b ≤ a + y , in view of which W (cid:129) a + b a + y ‹ = • a + b a + b a + b ˜ ⊔ W (cid:129) a + b a + y − b ‹ .Meanwhile of course W ( y , y ) = [ y ] , and (since y ≤ b ) W ( a + y − b , a ) =[ a + y − b ] ⊔ W ( a + y − b , b − y ) . So (3.2) amounts to the statement that • y , a + y − b , a + b a + b a + b ˜ ⊔W (cid:129) a + b a + y − b ‹ ⊔W ( b − y , a + y − b ) (cid:22) [ a + y ] .Applying Proposition 3.9 and reordering the sequence in brackets shows thatthis is equivalent to the statement that • a + b a + b b − a a − b b − y ˜ ⊔W (cid:129) a + b a + y − b ‹ ⊔W ( b − y , a + y − b ) (cid:22) • a + b ˜ .Then another application of Proposition 3.9 shows that this last statement (andhence also (3.2)) is equivalent to the statement that • a − b a − b a − b b − y ˜ ⊔W (cid:129) a + b a + y − b ‹ ⊔W ( b − y , a + y − b ) (cid:22) [ a ] .The left hand side above can be rewritten as [ ] ⊔W (cid:129) a − b a − b ‹ ⊔W (cid:129) a + b a + y − b ‹ ⊔W ( b − y , a + y − b ) ⊔W ( b − y , b − y ) . So by Theorem 3.2 it suffices to show that for all α > E (cid:129) a − b a − b ‹ ⊔ E (cid:129) a + b a + y − b ‹ ⊔ E ( b − y , a + y − b ) ⊔ E ( b − y , b − y ) , → α E ( a , a ) .We now repeatedly use Proposition 3.4, obtaining for any ν > • E ( b − y , a + y − b ) ⊔ E ( b − y , b − y ) , → ν E ( b − y , a ) ; • E (cid:0) a − b , 2 a − b (cid:1) ⊔ E (cid:0) a + b , a + y − b (cid:1) ⊂ E (cid:0) a − b , a + y − b (cid:1) ⊔ E (cid:0) a + b , a + y − b (cid:1) , → ν E ( a , a + y − b ) (since a ≤ y ); • E ( a , a + y − b ) ⊔ E ( b − y , a ) ∼ = E ( a + y − b , a ) ⊔ E ( b − y , a ) , → ν E ( a , a ) .Combining these embeddings (with ν = p α ) yields the embedding (3.3) andhence proves the proposition. (cid:3) Proof of Theorem 1.13.
We begin with the following easy observation,using the terminology and notation from Section 1.
Proposition 3.11.
Let Ω ⊂ [ ∞ ) n be any star-shaped domain such that X Ω contains the origin in its interior. Then(3.4) δ ell ( X Ω ) ≤ inf (cid:26) (cid:13)(cid:13)(cid:13)(cid:13)(cid:129) a , . . . , 1 a n ‹(cid:13)(cid:13)(cid:13)(cid:13) ∗ Ω (cid:12)(cid:12)(cid:12)(cid:12) There is a symplectic embedding E ( a , . . . , a n ) , → X ◦ Ω (cid:27) and(3.5) δ ell ( X Ω ) ≤ n [( a , . . . , a n )] Ω (cid:12)(cid:12)(cid:12) There is a symplectic embedding X Ω , → E ( a , . . . , a n ) ◦ o . Proof.
We first prove (3.4). Suppose that there is a symplectic embedding E ( a , . . . , a n ) , → X ◦ Ω and let α = k ( a , . . . , a n ) k ∗ Ω . So by definition, each point ( x , . . . , x n ) ∈ Ω obeys P i x i a i ≤ α . But α E ( a , . . . , a n ) is precisely the preimage under µ of (cid:8) ( x , . . . , x n ) ∈ [ ∞ ) n | P i x i a i ≤ α (cid:9) , while X Ω = µ − ( Ω ) . So we have E ( a , . . . , a n ) , → X ◦ Ω and X Ω ⊂ α E ( a , . . . , a n ) , and hence for E = α E ( a , . . . , a n ) there are sym-plectic embeddings X Ω , → E , → α X ◦ Ω . Thus δ ell ( X Ω ) ≤ α . Since ( a , . . . , a n ) was arbitrary subject to the assumption that there is a symplectic embedding E ( a , . . . , a n ) , → X ◦ Ω , this proves (3.4).Similarly, suppose that there is a symplectic embedding X Ω , → E ( a , . . . , a n ) ◦ and let ν = ” ( a , . . . , a n ) — Ω . Then for each ( x , . . . , x n ) ∈ [ ∞ ) \ Ω we have P i x i a i ≥ ν .So since Ω is closed it then follows that (cid:8) ( y , . . . , y n ) ∈ [ ∞ ) n | P i y i a i ≤ ν (cid:9) ⊂ Ω . Taking preimages under µ then shows that ν E ( a , . . . , a n ) ⊂ X Ω , andhence E ( a , . . . , a n ) , → ν − X Ω . Thus δ ell ( X Ω ) ≤ ν , which implies (3.5) since ν was arbitrary subject to the assumption that there is a symplectic embedding X Ω , → E ( a , . . . , a n ) ◦ . (cid:3) YMPLECTICALLY KNOTTED CODIMENSION-ZERO EMBEDDINGS OF DOMAINS IN R The proof of Theorem 1.13 now follows almost immediately based on Propo-sitions 3.5, 3.6, and 3.10. For part (a), the hypotheses that ˆ Ω is convex and that ( a , 0 ) , ( b ) , ( x , y ) ∈ Ω imply that also (
0, 0 ) , ( y ) ∈ Ω . Since ( y ) ∈ Ω andsince the right-hand-side of the desired inequality is independent of b , there isno loss of generality in assuming that b ≥ y , while the hypothesis of the theoremgives inequalities x ≤ a ≤ b ≤ x + y . The fact that Ω is a convex region con-taining ( a , 0 ) , (
0, 0 ) , ( b ) , ( x , y ) implies that the quadrilateral with these pointsas its vertices is contained in Ω , and hence that T ( a , b , x , y ) ⊂ X Ω . So for any α > E (cid:0) α − a , α − ( x + y ) (cid:1) , → X ◦ Ω ,whence (3.4) yields Theorem 1.13 (a).Similarly in part (b), by hypothesis we have ( a , 0 ) , ( x , y ) , ( b ) ∈ [ ∞ ) \ Ω ,and moreover [ ∞ ) \ Ω (and hence also its closure) is convex. Since Ω is bounded, it follows that [ ∞ ) \ Ω contains all points of form t ~ v where t ≥ ~ v lies on the line segment from ( a , 0 ) to ( x , y ) or the line segmentfrom ( x , y ) to ( b ) . The preimage under µ of the set of all such points is R \ T ( a , b , x , y ) ◦ , while the preimage under µ of [ ∞ ) \ Ω is R \ X ◦ Ω , sothis shows that X ◦ Ω ⊂ T ( a , b , x , y ) ◦ and hence (recalling our convention that“domains” are closures of open subsets) that X Ω ⊂ T ( a , b , x , y ) . Thus part (b)of Theorem 1.13 follows from Proposition 3.6 and (3.5).Part (c) of Theorem 1.13 is an immediate application of Proposition 3.10(applied to P ( a , b ) = T ( a , b , a , b ) ) together with (3.4).3.2. An explicit construction.
The embeddings from Propositions 3.5, 3.6,and 3.10 that underlie Theorem 1.13 are obtained by very indirect methodsand are difficult to understand concretely. We will now explain a more directconstruction that, for instance, leads to an explicit formula for a knotted em-bedding P (
1, 1 ) → α P (
1, 1 ) ◦ for any α ∈ € −p , 2 Š .The key ingredient is a toric structure on the complement of the antidiagonalin S × S that appears (at least implicitly) in [ EP09, Example 1.22 ] , [ FOOO12 ] , [ OU16, Section 2 ] . View S as the unit sphere in R and let A = { ( v , w ) ∈ S × S | w = − v } be the antidiagonal. Define functions F , F : S × S → R by F ( v , w ) = v + w F ( v , w ) = k v + w k .Now F fails to be smooth along A = F − ( { } ) , but on S × S \ A the Hamiltonianflows of the functions F and F induce S -actions that commute with eachother and are rather simple to understand: F induces simultaneous rotationof the factors about the z -axis, and F induces the flow which rotates the pair ( v , w ) ∈ S × S \ A about an axis in the direction of v + w . In formulas: φ tF (cid:0) ( v , v , v ) , ( w , w , w ) (cid:1) (3.6) = €(cid:0) ( cos t ) v − ( sin t ) v , ( sin t ) v + ( cos t ) v , v (cid:1) , (cid:0) ( cos t ) w − ( sin t ) w , ( sin t ) w + ( cos t ) w , w (cid:1)Š and(3.7) φ tF ( v , w ) = (cid:129) v + w + ( cos t ) v − w + ( sin t ) w × v k v + w k , v + w + ( cos t ) w − v + ( sin t ) v × w k v + w k ‹ .Define J : S × S → R by J ( v , w ) = ( − k v + w k , k v + w k − v − w ) , i.e. J = ( − F , − F + F ) . Then J is smooth away from A , and its restriction to S × S \ A is the moment map for a Hamiltonian T -action. It is not hard tosee that J has image equal to ∆ : = { ( x , y ) ∈ [ ∞ ) | x / + y / ≤ } , and thatthe preimage of { x / + y / = } is equal to Q : = { ( v , w ) ∈ S × S | v + w = −k v + w k} . (In other words, Q is the locus of pairs ( v , w ) ∈ S × S such that v + w is on the nonpositive z axis.) Proposition 3.12.
Let ∆ ◦ = (cid:8) ( x , y ) ∈ [ ∞ ) (cid:12)(cid:12) x + y < (cid:9) and define s : ∆ ◦ → S × S by s ( x , y ) = (cid:18)(cid:18)s x (cid:16) − x (cid:17) , s y (cid:16) − x − y (cid:17) , 1 − x + y (cid:19) , (cid:18) − s x (cid:16) − x (cid:17) , s y (cid:16) − x − y (cid:17) , 1 − x + y (cid:19)(cid:19) .Then, writing E ( π , 8 π ) ◦ = ¦ ( w , z ) ∈ C | | w | + | z | < © , the map Φ (cid:0) | z | e i θ , | z | e i ϕ (cid:1) = φ ϕ F (cid:18) φ θ − ϕ F (cid:18) s (cid:18) | z | | z | (cid:19)(cid:19)(cid:19) defines a symplectomorphism Φ : E ( π , 8 π ) ◦ → S × S \ Q which satisfies J ◦ Φ ( z , z ) = € | z | , | z | Š . Proof.
First we observe that s indeed takes values in S × S ⊂ R × R , whichfollows by computing x (cid:16) − x (cid:17) + y (cid:16) − x − y (cid:17) + (cid:16) − x + y (cid:17) = x + y − x + y − x y + − x − y + x + x y + y = ( x , y ) ∈ ∆ ◦ , if we write ( v , w ) = s ( x , y ) , then k v + w k = y (cid:16) − x − y (cid:17) + ( − x − y ) = x − x + = ( − x ) ,so (since x < J (cid:0) s ( x , y ) (cid:1) = (cid:0) − k v + w k , − v − w + k v + w k (cid:1) = ( x , x + y − + − x ) = ( x , y ) . Here we view T as ( R / π Z ) . On the other hand the map µ ( w , z ) = ( π | w | , π | z | ) thatwe have considered elsewhere is the moment map for a Hamiltonian ( R / Z ) -action; to get a ( R / π Z ) -action one would take µ π . YMPLECTICALLY KNOTTED CODIMENSION-ZERO EMBEDDINGS OF DOMAINS IN R In particular, the image of s is contained in S × S \ Q = J − ( ∆ ◦ ) , and it intersectseach fiber of J | J − ( ∆ ◦ ) just once.Moreover, since the image of s is contained in (cid:8) ( v , Rv ) | v ∈ S (cid:9) where R isthe reflection through the v v -plane and hence is antisymplectic, we see that s ⋆ Ω = Ω is the standard product symplectic form on S × S . Thus s : ∆ ◦ → J − ( ∆ ◦ ) is a Lagrangian right inverse to the moment map J .Write ψ ( θ , ϕ ) ( z , z ) = ( e − i θ z , e − i ϕ z ) for the standard T -action on E ( π , 8 π ) ◦ (with moment map µ π having image equal to ∆ ◦ ; the negative signs in front of θ and ϕ arise because our convention for Hamiltonian vector fields is ω ( X H , · ) = d H ). Likewise write ψ ( θ , ϕ ) = φ − ϕ F ◦ φ ϕ − θ F for the T -action on S × S \ Q in-duced by the moment map J . Our map Φ maps the Lagrangian section of µ π given by the nonnegative real locus of E ( π , 8 π ) ◦ to the Lagrangian section of J | S × S \ Q given by the image of s , and Φ obeys J ◦ Φ = µ π and, for all ( θ , ϕ ) ∈ T , Φ ◦ ψ ( θ , ϕ ) = ψ ( θ , ϕ ) ◦ Φ . These facts are easily seen to imply that Φ is a sym-plectomorphism, as it identifies action-angle coordinates on E ( π , 8 π ) ◦ withaction-angle coordinates on S × S \ Q . The last statement is immediate fromthe formula for Φ and the facts that J ◦ s is the identity and that J is preservedunder the Hamiltonian flows of F and F . (cid:3) Remark . With sufficient effort, one can derive the following equivalentformula for the map Φ : E ( π , 8 π ) ◦ → S × S from Proposition 3.12: regarding S as the unit sphere in C × R , we have Φ ( w , z ) = (cid:0) Γ ( w , z ) , Γ ( − w , z ) (cid:1) where Γ ( w , z ) = ‚ p − | w | (cid:0) ( − | w | − | z | ) w + ¯ wz (cid:1) ( − | w | ) + iz Æ − | w | − | z | ,(3.8) 1 − | w | + | z | − p ( − | w | )( − | w | − | z | ) ( − | w | ) I m ( w ¯ z ) (cid:19) .Since E ( π , 8 π ) ◦ is precisely the locus where 2 | w | + | z | <
8, this makes clearthat Φ is a smooth (indeed even real-analytic) map despite the appearance ofsquare roots in the formula for s in Proposition 3.12.Now if D ( π ) denotes the open disk of area 4 π (so radius 2) in C , there is asymplectomorphism σ : S \ { ( − ) } → D ( π ) defined by(3.9) σ ( z , v ) = vt + v z where as in Remark 3.13 we regard S as the unit sphere in C × R .So if we let I = (cid:0) { ( − ) } × S (cid:1) ∪ (cid:0) S × { ( − ) } (cid:1) then σ × σ defines asymplectomorphism S × S \ I ∼ = P ( π , 4 π ) ◦ = D ( π ) × D ( π ) .For v = ( z , v ) ∈ S ⊂ C × R , we have k v + ( − ) k = | z | + v − v + = − v
36 JEAN GUTT AND MICHAEL USHER and hence J (cid:0) v , ( − ) (cid:1) = J (cid:0) ( − ) , v (cid:1) = (cid:0) − p − v , p − v + ( − v ) (cid:1) .Thus J ( I ) ⊂ (cid:8) ( x , y ) ∈ R | ( − x ) = ( x + y ) − (cid:9) = (cid:8) ( x , y ) ∈ R | y = x − x + (cid:9) .Since µ π = J ◦ Φ , we have µ π ( Φ − ( I )) = J ( I ) . From this we obtain the follow-ing: Proposition 3.14.
Suppose that X Ω is a convex toric domain where Ω ⊂ (cid:8) ( π x , 2 π y ) ∈ [ ∞ ) | y < x − x + (cid:9) . Then there is an ellipsoid E such that X Ω ⊂ E ◦ andsuch that the map Φ from Proposition 3.12 maps E to a subset of S × S \ I .Hence ( σ × σ ) ◦ Φ | E is a symplectic embedding from E to P ( π , 4 π ) ◦ . Proof.
The sets π Ω and S : = (cid:8) ( x , y ) ∈ [ ∞ ) | y ≥ x − x + (cid:9) are dis-joint, closed, convex subsets of R , and the first of these sets is compact, sothe hyperplane separation theorem shows that they must be separated by a line ℓ , which passes through the first quadrant since both sets are contained in thefirst quadrant. This line ℓ must have negative slope, since S intersects all lineswith positive slope and also intersects all horizontal or vertical lines that passthrough the first quadrant. So we can write the separating line as ℓ = (cid:8) ( x , y ) ∈ R | xa + yb = (cid:9) with a , b >
0, and then it will hold that π Ω ⊂ (cid:8) xa + yb < (cid:9) and S ⊂ { xa + yb > } . The first inclusion shows that X Ω ⊂ E ( π a , 2 π b ) ◦ . Mean-while since (
2, 0 ) , (
0, 4 ) ∈ S ⊂ (cid:8) xa + yb > (cid:9) , we have a < b < E ( π a , 2 π b ) is contained in the domain of the map Φ from Proposition3.12, and by the discussion before the proposition the fact that ℓ ∩ S = ∅ implies that E ( π a , 2 π b ) ∩ Φ − ( I ) = ∅ . Hence the proposition holds with E = E ( π a , 2 π b ) . (cid:3) Corollary 3.15.
Suppose that X Ω is a convex toric domain with Ω ⊂ { ( π x , 2 π y ) ∈ [ ∞ ) | y < x − x + } , and that we have P ( π , 4 π ) ⊂ α X Ω for some α <δ uell ( X Ω ) . Then ( σ × σ ) ◦ Φ | X Ω : X Ω , → P ( π , 4 π ) ◦ ⊂ α X ◦ Ω defines a knottedembedding X Ω , → α X ◦ Ω . Proof.
By Proposition 3.14 we have an ellipsoid E and a sequence X Ω , → E ◦ , → P ( π , 4 π ) ◦ ⊂ α X ◦ Ω where the first map is the inclusion and the second mapis ( σ × σ ) ◦ Φ | E . So the corollary follows directly from the assumption that α < δ uell ( X Ω ) and the definition of δ uell . (cid:3) We emphasize that this embedding ( σ × σ ) ◦ Φ is completely explicit: σ is defined in (3.9) and Φ is defined in Proposition 3.12 based partly on theformulas (3.6) and (3.7), or even more explicitly is given by (3.8). Example 3.16.
For instance, Ω could be taken to be a square [
0, 2 π c ] with c smaller than the smallest root of the polynomial x − x +
4, namely 4 − p YMPLECTICALLY KNOTTED CODIMENSION-ZERO EMBEDDINGS OF DOMAINS IN R F IGURE
4. After appropriate rescalings, the map Φ from Propo-sition 3.12 sends the interior of the ellipsoid E (
1, 2 ) to a productof spheres of area 1, with the preimage of ( S × { (
0, 0, − ) } ) ∪ ( { (
0, 0, − ) } × S ) contained in µ − ( C ) where C is the red curveat left. Consequently the preimage under µ of any domain lyingbelow C , such as the small square at left, is embedded into thepolydisk P (
1, 1 ) ◦ by a rescaling of ( σ × σ ) ◦ Φ . This gives anexplicit knotted embedding P ( c , c ) , → P (
1, 1 ) ◦ for 1 / < c < − p ( σ × σ ) ◦ Φ : P ( π c , 2 π c ) , → P ( π , 4 π ) ◦ = c P ( π c , 2 π c ) ◦ , which is knotted provided that c < δ uell (cid:0) P ( π c , 2 π c ) (cid:1) .By Theorem 1.12 we have δ uell (cid:0) P ( a , a ) (cid:1) ≥ a , so our embedding is knot-ted provided that 1 < c < − p
2. So after conjugating by appropriate rescal-ings our explicit embedding ( σ × σ ) ◦ Φ defines a knotted embedding P ( a , a ) , → α P ( a , a ) ◦ provided that 2 > α > −p ≈ δ ell ( P ( a , a )) ≤ / P ( a , a ) , → α P ( a , a ) ◦ whenever2 > α > P ( π , 4 π ) ◦ , the image of thisembedding α − P ( π , 4 π ) , → P ( π , 4 π ) ◦ is not hard to describe explicitly as asubset of P ( π , 4 π ) ◦ : it is given as the region (cid:8) ( z , z ) ∈ P ( π , 4 π ) ◦ | G ( z , z ) ≥ − /α , − G ( z , z ) + G ( z , z ) ≤ /α (cid:9) ,where G i = F i ◦ ( σ × σ ) − , i.e. , G ( z , z ) = − | z | + | z | and G ( z , z ) = ‚vt − | z | ( z ) + vt − | z | ( z ) Œ + ‚vt − | z | ( z ) + vt − | z | ( z ) Œ + (cid:18) − | z | + | z | (cid:19) .Corollary 3.15 also applies to some other convex toric domains besides thecube P ( a , a ) , though it as not as broadly applicable as Theorem 1.8. For examplethe reader may check that, in Corollary 3.15, for appropriate α one can take X Ω equal to a polydisk P ( a ) with 1 ≤ a ≤ ℓ p ball as in Theorem 1.8 for p ≥ Remark . By construction, the embedding Φ from Proposition 3.12 mapsthe torus T p : = (cid:8) ( w , z ) ∈ C (cid:12)(cid:12) | w | = | z | = p (cid:9) to the Lagrangian torus in S × S that is denoted K in [ EP09, Example 1.22 ] , and which can be identifiedwith the Chekanov-Schlenk twist torus Θ , see [ CS10 ] , [ OU16 ] . Since, as shownin [ EP09 ] , there is no symplectomorphism mapping K to the Clifford torus in S × S ( i.e. , to the image of T p under the standard embedding ( σ × σ ) − of P ( π , 4 π ) ◦ into S × S ), one easily infers independently of our other results that ( σ × σ ) ◦ Φ : P ( π c , 2 π c ) , → P ( π , 4 π ) ◦ must not be isotopic to the inclusionby a compactly supported Hamiltonian isotopy for 1 < c < − p S × S , giving a symplectomorphismthat would send K to the Clifford torus). However this argument based onLagrangian tori does not seem to adapt to yield the full result that ( σ × σ ) ◦ Φ is knotted in the stronger sense of Definition 1.7.By the way, if c <
1, our embedding ( σ × σ ) ◦ Φ : P ( π c , 2 π c ) , → P ( π , 4 π ) ◦ isunknotted. Indeed in this case the ball B ( π c ) is contained both in P ( π , 4 π ) ◦ and in E ( π , 8 π ) \ Φ − ( I ) , and so both ( σ × σ ) ◦ Φ | P ( π c ,2 π c ) and the inclusion P ( π c , 2 π c ) , → P ( π , 4 π ) ◦ extend to embeddings B ( π c , 4 π c ) , → P ( π , 4 π ) ◦ ;these two embeddings of the ball are symplectically isotopic by [ C14, Proposi-tion 1.5 ] . Thus a transition between knottedness and unknottedness occurs atthe value c =
1, which is precisely the first value for which P ( π c , 2 π c ) containsthe torus T p mentioned at the start of the remark. Remark . A similar construction to that in Proposition 3.12, using resultsfrom [ OU16, Section 3 ] , allows one to construct a symplectic embedding of E ( π , 12 π ) ◦ into C P where the symplectic form on C P is normalized to givearea 6 π to a complex projective line, such that the torus T p is sent to the C P version of the Chekanov-Schlenk twist torus Θ . Combining this with a sym-plectomorphism from the complement of a line in C P to a ball and restrictingto P ( π c , 2 π c ) for c slightly larger than 1, we obtain a symplectic embedding P ( π c , 2 π c ) , → B ( π ) ◦ which cannot be Hamiltonian isotopic to the inclu-sion because Θ is not Hamiltonian isotopic to the Clifford torus. It is less clearwhether this embedding P ( π c , 2 π c ) , → B ( π ) ◦ is knotted in the sense of Def-inition 1.7; the symplectic-homology-based methods in the present paper seem YMPLECTICALLY KNOTTED CODIMENSION-ZERO EMBEDDINGS OF DOMAINS IN R ill-equipped to address this because the filtered positive S -equivariant symplec-tic homology of B ( π ) does not have as rich a structure as that of the domains X that appear in Theorem 1.8.4. M ORE KNOTTED POLYDISKS
The lower bounds on δ uell that are used to show that our embeddings X , → α X ◦ are knotted are generally based on showing that, for suitable k , L , the maps Φ L : C H Lk ( α X ◦ , λ ) → C H Lk ( X ◦ , λ ) have sufficiently large image and then ap-pealing to Corollary 2.19. One can in principle use Corollary 2.19 to provethe knottedness of embeddings X , → V for more general star-shaped open sub-sets V which are not dilates of X ◦ ; the main difficulty in this case is that onecan no longer simply appeal to Lemma 2.11 in order to estimate the rank of Φ L : C H Lk ( V , λ ) → C H Lk ( X ◦ , λ ) .In this section we carry this procedure out when X and V are four-dimensionalpolydisks P ( a , b ) , P ( a , b ) ◦ , typically with b a = b a .A polydisk P ( a , b ) is the toric domain associated to the rectangle R a , b =[ a ] × [ b ] , which has k ( x , y ) k ∗ R a , b = a x + b y , so by Lemma 2.12 we seethat dim C H L (cid:0) P ( a , b ) ◦ , λ (cid:1) = { a , b } < L < a + b , and that ı L , L : C H L (cid:0) P ( a , b ) ◦ , λ (cid:1) → C H L (cid:0) P ( a , b ) ◦ , λ (cid:1) is an isomorphism for(4.1) max { a , b } < L < L < a + b . Lemma 4.1.
Assume that a ≤ b < b ′ and that b ′ < L < a + b . Then the transfermap Φ L : C H L (cid:0) P ( a , b ′ ) ◦ , λ (cid:1) → C H L (cid:0) P ( a , b ) ◦ , λ (cid:1) is an isomorphism. Proof.
Choose δ > ( + δ ) L < a + b and such that, for some N ∈ N , ( + δ ) N b = b ′ . Consider any c ∈ [ b , b ′ ] . Lemma 2.11 gives a commutativediagram C H L (cid:0) ( + δ ) − P ( a , c ) ◦ , λ (cid:1) ∼ = (cid:15) (cid:15) C H L (cid:0) P ( a , c ) ◦ , λ (cid:1) Φ L ❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ ı L , ( + δ ) L * * ❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚ C H ( + δ ) L (cid:0) P ( a , c ) ◦ , λ (cid:1) where ı L , ( + δ ) L is an isomorphism by Lemma 2.12 since our assumptions give a ≤ b ≤ c ≤ b ′ < L < ( + δ ) L < a + b ≤ a + c . Thus Φ L : C H L (cid:0) P ( a , c ) ◦ , λ (cid:1) → C H L € P (cid:0) ( + δ ) − a , ( + δ ) − c (cid:1) , λ Š is an isomorphism. But this latter mapfactors as a composition C H L (cid:0) P ( a , c ) ◦ , λ (cid:1) Φ L / / C H L (cid:0) P ( a , ( + δ ) − c ) ◦ , λ (cid:1) Φ L / / C H L € P (cid:0) ( + δ ) − a , ( + δ ) − c (cid:1) ◦ , λ Š , so since all three vector spaces above have dimension two it follows that, againfor any c ∈ [ b , b ′ ] , Φ L : C H L (cid:0) P ( a , c ) ◦ , λ (cid:1) → C H L (cid:0) P ( a , ( + δ ) − c ) , λ (cid:1) is anisomorphism.Since ( + δ ) − N b ′ = b , we may apply this successively with c = b ′ , ( + δ ) − b ′ , . . . , ( + δ ) − ( N − ) b ′ and appeal to the functoriality of Φ L to see that Φ L : C H L (cid:0) P ( a , b ′ ) ◦ , λ (cid:1) → C H L (cid:0) P ( a , b ) ◦ , λ (cid:1) is an isomorphism. (cid:3) A similar argument gives:
Lemma 4.2.
Assume that a < a ′ ≤ b and that b < L < a + b . Then the transfermap Φ L : C H L (cid:0) P ( a ′ , b ) ◦ , λ (cid:1) → C H L (cid:0) P ( a , b ) ◦ , λ (cid:1) is an isomorphism. Proof.
Analogously to the proof of Lemma 4.1, choose δ > ( + δ ) L < a + b and, for some N ∈ N , ( + δ ) N a = a ′ . Using Lemma 2.11 and(4.1), we find that for all c ∈ [ a , a ′ ] the transfer map Φ L : C H L (cid:0) P ( c , b ) ◦ , λ (cid:1) → C H L € P (cid:0) ( + δ ) − c , ( + δ ) − b (cid:1) ◦ , λ Š is an isomorphism. Since this map factorsthrough C H L (cid:0) P ( + δ ) − c , b ) , λ (cid:1) , we deduce by dimensional considerationsthat Φ L : C H L (cid:0) P ( c , b ) , λ (cid:1) → C H L (cid:0) P (( + δ ) − c , b ) , λ (cid:1) is an isomorphism forall c ∈ [ a , a ′ ] . Just as in the proof of Lemma 4.1, iterating this for c = a ′ , ( + δ ) − a ′ , . . . , ( + δ ) − ( N − ) a ′ yields the result. (cid:3) Proposition 4.3.
Assume that a ≤ b , that a ≤ a and that b ≤ b . Forany L with b < L < a + b , the transfer map Φ L : C H L (cid:0) P ( a , b ) ◦ , λ (cid:1) → C H L (cid:0) P ( a , b ) ◦ , λ (cid:1) is an isomorphism (and so in particular has rank two). Proof. If a ≤ b and b a ≥ b a , then we can factor Φ L : C H L ( P ( a , b ) ◦ ) → C H L ( P ( a , b ) ◦ ) as a composition C H L ( P ( a , b ) ◦ ) Φ L / / C H L € P € a , a b a Š ◦ Š Φ L / / C H L ( P ( a , b ) ◦ ) where the first map is an isomorphism by Lemma 4.1 and the second is anisomorphism by (4.1) and Lemma 2.11 (which identifies the map with theinclusion-induced map ı a a L , L : C H a a L (cid:0) P ( a , b ) ◦ , λ (cid:1) → C H L (cid:0) P ( a , b ) ◦ , λ (cid:1) ).Similarly, if a ≤ b and b a ≤ b a , then we can factor Φ L : C H L (cid:0) P ( a , b ) ◦ , λ (cid:1) → C H L (cid:0) P ( a , b ) ◦ , λ (cid:1) as a composition C H L (cid:0) P ( a , b ) ◦ , λ (cid:1) Φ L / / C H L € P € a b b , b Š ◦ , λ Š Φ L / / C H L (cid:0) P ( a , b ) ◦ , λ (cid:1) where the first map is an isomorphism by Lemma 4.2 and the second is anisomorphism by (4.1) and Lemma 2.11.We have now proven the result whenever a ≤ b . If instead a > b , thenthe hypotheses imply that P ( a , b ) ⊂ P ( a , a ) ⊂ P ( a , b ) , and that a ≤ b < L < a + b < a . We can then factor the map in question as C H L (cid:0) P ( a , b ) ◦ , λ (cid:1) Φ L / / C H L (cid:0) P ( a , a ) ◦ , λ (cid:1) Φ L / / C H L (cid:0) P ( a , b ) ◦ , λ (cid:1) YMPLECTICALLY KNOTTED CODIMENSION-ZERO EMBEDDINGS OF DOMAINS IN R where the first map is an isomorphism by a case of the present corollary that wehave already proven, and the second is an isomorphism by Lemma 4.2 (afterconjugating by a symplectomorphism that switches the factors of C ). (cid:3) Corollary 4.4. If E ⊂ R is an ellipsoid, if g : E → P ( c , d ) ◦ is a symplecticembedding where c ≤ d , and if P ( a , b ) ⊂ E ∩ P ( c , d ) ◦ , then g | P ( a , b ) is knottedprovided that d < a + b . Proof. If g | P ( a , b ) were unknotted, we could apply Corollary 2.19 with f equal tothe inclusion, with k =
3, and with L equal to any number with d < L < a + b .This would yield Rank € Φ L : C H L (cid:0) P ( c , d ) ◦ , λ (cid:1) → C H L (cid:0) P ( a , b ) ◦ , λ (cid:1)Š ≤
1, incontradiction with Proposition 4.3. (cid:3)
Remark . In the case that max { a , b } < c , so that P ( c , d ) ◦ contains both P ( a , b ) and P ( b , a ) , then one example of a symplectic embedding P ( a , b ) , → P ( c , d ) ◦ is σ : ( z , z ) ( z , z ) , which has image equal to P ( b , a ) . In [ FHW94,Theorem 4 ] it is shown that, when c = d < a + b , this embedding is not Hamil-tonian isotopic to the inclusion within P ( c , c ) ◦ . However our definition of knot-tedness is such that (when c = d ) this embedding would be considered unknot-ted, because the symplectomorphism of P ( c , c ) ◦ which swaps the factors maps P ( a , b ) to P ( b , a ) (and we do not require our ambient symplectomorphismsto be induced by Hamiltonian isotopies supported in the codomain). Likewisewhen a = b but c = d , σ is unknotted according to our definition because wetake knottedness to depend only on the image of the embedding.In the situation that both a = b and c < d (and still max { a , b } ≤ c and d < a + b ) it can be shown that the above embedding σ : P ( a , b ) → P ( c , d ) with image P ( b , a ) is knotted. More specifically, by using arguments like those in [ FHW94,Section 3.3 ] one can show that for a < L < b and c < L < d the inclusion-induced map SH [ L , L ) ( P ( c , d ) ◦ ) → SH [ L , L ) ( P ( a , b ) ◦ ) on action-window sym-plectic homology vanishes, while the inclusion-induced map SH [ L , L ) ( P ( c , d ) ◦ ) → SH [ L , L ) ( P ( b , a ) ◦ ) is nontrivial, which is sufficient to show that P ( a , b ) cannotbe mapped to P ( b , a ) by a symplectomorphism of P ( c , d ) ◦ ; we omit the details.However because Proposition 4.3 shows that, for d < L < a + b , the map Φ L : C H L (cid:0) P ( c , d ) ◦ , λ (cid:1) → C H L (cid:0) P ( b , a ) ◦ , λ (cid:1) has rank two, the embeddings de-scribed by Corollary 4.4 (for which Φ L has rank one) have different knot typesfrom σ . (In other words, the image of such an embedding is not taken by asymplectomorphism of P ( c , d ) ◦ to either one of P ( a , b ) or P ( b , a ) .) In particu-lar this comment applies to the embeddings in Corollaries 4.6 and 4.7 in eachcase that the target contains the image of the domain under σ . Corollary 4.6.
Let m ∈ Z + and 0 ≤ ǫ <
1. If a + b m + ǫ < ≤ a ≤ b < m + ǫ < a + b then there is a knotted embedding of P ( a , b ) into P ( m + ǫ ) ◦ . Proof.
Choose µ such that a + b m + ǫ < µ <
1; we then have P ( a , b ) ⊂ ( µ E (
1, 2 m + ǫ )) ∩ P ( m + ǫ ) ◦ . Proposition 3.8 moreover gives a symplectic embedding µ E (
1, 2 m + ǫ ) , → P ( m + ǫ ) ◦ . The conclusion then follows from Corollary4.4. (cid:3) We conclude by restating and proving Theorem 1.10:
Corollary 4.7.
Given any y ≥
1, there exist polydisks P ( a , b ) and P ( c , d ) andknotted embeddings of P ( a , b ) into P ( y ) ◦ and of P ( y ) into P ( c , d ) ◦ . Proof.
For a knotted embedding P ( a , b ) , → P ( y ) ◦ , write y = m + ǫ where m ∈ Z + and 0 ≤ ǫ <
1. We can then set a = and b = m + ǫ − and applyCorollary 4.6.For a knotted embedding P ( y ) , → P ( c , d ) ◦ , write y = k + δ where k ∈ Z + and − ≤ δ <
1. If δ ≥
0, then Corollary 4.6 gives a knotted embedding of P ( µ , µ y ) into P (cid:0) k + δ (cid:1) ◦ for any µ with12 2 k + δ k + δ + < µ < k + δ k + δ ,and so conjugating by a rescaling by µ gives the desired embedding (with c = µ , d = µ (cid:0) k + δ (cid:1) = y µ ). If instead − ≤ δ <
0, then for 1 + δ k < α < + + δ k Corollary 4.6 (with m = k , ǫ =
0) gives a knotted embedding of α P ( y ) into P ( k ) ◦ , and so again conjugating by a rescaling gives the desired embeddingwith c = α , d = α k . (cid:3) A PPENDIX
A. P
ROOF OF L EMMA [ GG16 ] for a slightly different version of S -equivariant symplectichomology; the main difference between our result and theirs is that they con-struct a filtered complex after choosing a certain action interval and prove thattheir complex computes the filtered S -equivariant symplectic homology asso-ciated to this action interval, whereas we construct a single complex that workssimultaneously for all action intervals. One can in fact show based on argu-ments similar to those below that the filtration on the complex constructed in [ GG16 ] in the case of the action interval ( ∞ ) does have filtered homologiesthat recover their version of filtered C H in arbitrary action intervals, but sincethis is not explicitly proven in [ GG16 ] we give a detailed proof in our case.The main ingredient is an algebraic lemma concerning filtered complexeswhich shows that, up to isomorphism, the images of inclusion-induced mapsbetween the filtered parts of the complexes can be recovered from the filteredhomology of a new chain complex whose underlying vector space is the E termof the spectral sequence associated to the original filtered complex. This lemmais proven in the following section, and in the subsequent section we apply thistogether with results from [ G15 ] , [ GH17 ] to complete the proof of Lemma 2.1.We assume that the reader is familiar with positive S -equivariant symplectichomology and we use the notation from [ GH17 ] .A.1. A lemma on filtered complexes.
In this section we consider a Z -gradedchain complex ( C ∗ , ∂ ) of vector spaces over a field K equipped with a filtration { } = F C ∗ ⊂ F C ∗ ⊂ · · · ⊂ F r C ∗ ⊂ · · · ⊂ C ∗ YMPLECTICALLY KNOTTED CODIMENSION-ZERO EMBEDDINGS OF DOMAINS IN R (where each F r C ∗ is a subcomplex of C ∗ ) that is bounded below by zero andexhausting ( i.e. F ∞ C ∗ : = ∪ r F r C ∗ is equal to C ∗ ). We extend the above filtrationby N to a filtration by Z by setting F i C ∗ = { } for i < ( C ∗ , ∂ ) , denoted G ( C ∗ ) , is thedirect sum of quotient complexes L p ≥ F p C ∗ F p − C ∗ , equipped with obvious boundaryoperator induced from ∂ . The homology H ∗ (cid:0) G ( C ∗ ) (cid:1) evidently splits as a directsum H k (cid:0) G ( C ∗ ) (cid:1) = M p ≥ H k (cid:18) F p C ∗ F p − C ∗ (cid:19) .The following is the main algebraic input needed for Lemma 2.1: Lemma A.1.
With notation and assumptions as above, there is a chain complex ( D ∗ , δ ) equipped with a filtration { } = F D ∗ ⊂ F D ∗ ⊂ · · · ⊂ F r D ∗ ⊂ · · · ⊂ D ∗ where for each r , k (A.1) F r D k = M ≤ p ≤ r H k (cid:18) F p C ∗ F p − C ∗ (cid:19) and F ∞ D ∗ : = ∪ r F r D ∗ = D ∗ , such that the boundary operator δ on D ∗ strictlylowers filtration in the sense that δ ( F r D ∗ ) ⊂ F r − D ∗ , and such that for 1 ≤ s ≤ t ≤ ∞ there exists an isomorphism of vector spacesIm (cid:0) H k ( F s C ∗ , ∂ ) → H k ( F t C ∗ , ∂ ) (cid:1) ∼ = Im (cid:0) H k ( F s D ∗ , δ ) → H k ( F t D ∗ , δ ) (cid:1) where the maps on both sides are induced by inclusion of filtered subcomplexes.The proof of Lemma A.1 will occupy the rest of this section. To begin, letus recall from [ Wei94, Section 5.4 ] some ingredients in the construction of thespectral sequence associated to the filtration on ( C ∗ , ∂ ) .For p ∈ Z write η p : F p C ∗ → F p C ∗ F p − C ∗ for the natural projection, and for p , q , r ∈ Z define: A rp , q = (cid:8) x ∈ F p C p + q | ∂ x ∈ F p − r C p + q − (cid:9) ,ˆ Z rp , q = η p ( A rp , q ) , ˆ B rp , q = η p € ∂ ( A r − p + r − q − r + ) Š .For any r ≥ { } = ˆ B p , q ⊂ ˆ B p , q ⊂ · · · ⊂ ˆ B rp , q ⊂ ˆ B r + p , q ⊂ ˆ Z r + p , q ⊂ ˆ Z rp , q ⊂ · · · ⊂ ˆ Z p , q = F p C p + q F p − C p + q .We also write ˆ B ∞ p , q = ∪ ∞ r = ˆ B rp , q = ∪ ∞ r = η p € ∂ ( A r − p + r − q − r + ) Š .Note also that since we assume that F i C ∗ = { } for i ≤
0, we haveˆ Z rp , q = ˆ Z pp , q = η p € ker ∂ | F p C p + q Š for r ≥ p . Accordingly if we let ˆ Z ∞ p , q = ˆ Z pp , q then we will haveˆ Z ∞ p , q = ∩ ∞ r = ˆ Z rp , q .As is standard, we write E rp , q = ˆ Z rp , q ˆ B rp , q for r ∈ N ∪ {∞} . For the case that r =
1, notice that ˆ Z p , q is equal to the set ofdegree- ( p + q ) cycles in the quotient complex F p C ∗ F p − C ∗ and that ˆ B p , q is equal to theset of degree- ( p + q ) boundaries in F p C ∗ F p − C ∗ ; thus(A.2) E p , q = H p + q (cid:18) F p C ∗ F p − C ∗ (cid:19) .The following is standard and easily-checked: Proposition A.2. (cf. [ Wei94, Construction 5.4.6 ] ) For each p , q , r , the bound-ary operator ∂ induces a mapˆ ∂ rp , q : E rp , q → E rp − r , q + r − such that ker ( ˆ ∂ rp , q ) = π ( ˆ Z r + p , q ) and Im ( ˆ ∂ rp + r , q − r + ) = π ( ˆ B r + p , q ) ,where π : ˆ Z rp , q → ˆ Z rp , q ˆ B rp , q is the quotient projection.We also have the following fact concerning the maps H k ( F p C ∗ , ∂ ) → H k ( F t C ∗ , ∂ ) for p ≤ t induced by inclusion of filtered subcomplexes; this is a slight exten-sion of the familiar fact that the spectral sequence of a suitable filtered complexconverges to the associated graded of the homology. Proposition A.3.
Let 1 ≤ p ≤ t ≤ ∞ with p < ∞ . Then there is an isomor-phism Im (cid:0) H k ( F p C ∗ , ∂ ) → H k ( F t C ∗ , ∂ ) (cid:1) Im (cid:0) H k ( F p − C ∗ , ∂ ) → H k ( F t C ∗ , ∂ ) (cid:1) ∼ = ˆ Z ∞ p , k − p ˆ B t − p + p , k − p .(Here for the case t = ∞ we interpret F ∞ C ∗ as C ∗ and ˆ B ∞− p + p , k − p as ˆ B ∞ p , k − p .) Proof.
There is an obvious surjective map φ : ker ( ∂ | F p C k ) → Im (cid:0) H k ( F p C ∗ , ∂ ) → H k ( F t C ∗ , ∂ ) (cid:1) Im (cid:0) H k ( F p − C ∗ , ∂ ) → H k ( F t C ∗ , ∂ ) (cid:1) given by including ker ( ∂ | F p C k ) into ker ( ∂ | F t C k ) , then taking homology classes,and then projecting. We see that x ∈ ker ( φ ) if and only if there is y ∈ ker ( ∂ | F p − C k ) such that x and y represent the same homology class in H k ( F t C ∗ , ∂ ) ; this holdsif and only if we can write x = y + ∂ z with z ∈ F t C k + , and in this case we YMPLECTICALLY KNOTTED CODIMENSION-ZERO EMBEDDINGS OF DOMAINS IN R would have z ∈ A t − pt , k − t + since ∂ z = x − y ∈ F p C k . Thus ker ( φ ) = ker ( ∂ | F p − C k )+ ∂ ( A t − pt , k − t + ) and hence(A.3) Im (cid:0) H k ( F p C ∗ , ∂ ) → H k ( F t C ∗ , ∂ ) (cid:1) Im (cid:0) H k ( F p − C ∗ , ∂ ) → H k ( F t C ∗ , ∂ ) (cid:1) ∼ = ker ( ∂ | F p C k ) ker ( ∂ | F p − C k ) + ∂ ( A t − pt , k − t + ) .(The above discussion implicitly assumes that t < ∞ , but since ∪ ∞ s = F s C ∗ = C ∗ the reasoning is equally valid for t = ∞ provided that we interpret the notation A t − p ∞ , k −∞ + as ∪ p ≤ s ∈ N A s − ps , k − s + , as we will continue to do below).On the other hand the projection η p : F p C k → F p C k F p − C k sends ker ( ∂ | F p C k ) toˆ Z ∞ p , k − p and sends ker ( ∂ | F p − C k ) + ∂ ( A t − pt , k − t + ) to ˆ B t − p + p , k − p , and it is easy to checkthat the resulting map η : ker ( ∂ | F p C k ) ker ( ∂ | F p − C k ) + ∂ ( A t − pt , k − t + ) → ˆ Z ∞ p , k − p ˆ B t − p + p , k − p is an isomorphism. Combining this isomorphism with (A.3) proves the propo-sition. (cid:3) For 1 ≤ r ≤ ∞ let B rp , q = ˆ B rp , q ˆ B p , q Z rp , q = ˆ Z rp , q ˆ B p , q ,so for r < p we have a chain of inclusions { } = B p , q ⊂ · · · ⊂ B rp , q ⊂ B r + p , q ⊂ · · · ⊂ B ∞ p , q ⊂ Z ∞ p , q = Z pp , q ⊂ Z r + p , q ⊂ Z rp , q ⊂ · · · ⊂ Z p , q = E p , q .Projecting away ˆ B p , q induces isomorphisms E rp , q ∼ = Z rp , q B rp , q . For each p , q ∈ Z and r ≥ • A complement H rp , q to the subspace B rp , q within the vector space Z rp , q ,and • A complement M rp , q to the subspace Z r + p , q within the vector space Z rp , q .Given these choices, the projection Z rp , q → E rp , q restricts to H rp , q as an isomor-phism, so the maps ˆ ∂ rp , q from Proposition A.2 induce maps ∂ rp , q : H rp , q → H rp − r , q + r − with ker ∂ rp , q = Z r + p , q ∩ H rp , q , Im ∂ rp + r , q − r + = B r + p , q ∩ H rp , q .(In particular, since Z r + p , q = Z rp , q for r ≥ p , we have ∂ rp , q = r ≥ p ).For any r ≥ Z j − p , q = Z jp , q ⊕ M j − p , q yielda direct sum decomposition E p , q = Z p , q = Z rp , q ⊕ M r − p , q ⊕ · · · ⊕ M p , q = H rp , q ⊕ B rp , q ⊕ M r − p , q ⊕ · · · ⊕ M p , q . (For r = B p , q = { } and H p , q = E p , q and the above direct sumdecomposition degenerates to E p , q = H p , q ).We accordingly extend our map ∂ rp , q : H rp , q → H rp , q to a linear map (still de-noted ∂ rp , q ) defined on all of E p , q by setting it equal to zero on the summands B rp , q , M r − p , q , . . . , M p , q . We also regard the codomain of ∂ rp , q as E p − r , q + r − ratherthan the subspace H rp − r , q + r − . With this extended definition, we haveker ∂ rp , q = ( Z r + p , q ∩ H rp , q ) ⊕ B rp , q ⊕ M r − p , q ⊕ · · · ⊕ M p , q = Z r + p , q ⊕ M r − p , q ⊕ · · · ⊕ M p , q ,where we have used that B rp , q ⊂ Z r + p , q ⊂ Z rp , q = H rp , q ⊕ B rp , q , so that ( Z r + p , q ∩ H rp , q ) ⊕ B rp , q = Z r + p , q . Since we have a direct sum decomposition E p , q = Z r + p , q ⊕ M rp , q ⊕ M r − p , q · · · ⊕ M p , q ,it follows that: Corollary A.4.
The maps ∂ rp , q : E p , q → E p − r , q + r − restrict as isomorphisms M rp , q → B r + p − r , q + r − ∩ H rp − r , q + r − , and vanish identically on the complementary subspace Z r + p , q ⊕ M r − p , q ⊕ · · · ⊕ M p , q to M rp , q in E p , q .In particular, since for j > r we have M jp , q ⊂ Z jp , q ⊂ Z r + p , q ⊂ ker ( ∂ rp , q ) , thisshows that ∂ rp , q vanishes on M jp , q for j = r , while it maps M rp , q isomorphically to B r + p − r , q + r − ∩ H rp − r , q + r − .Now for any p , q let us write ∂ p , q = X r ≥ ∂ rp , q : E p , q → ⊕ r ≥ H rp − r , q + r − ⊂ E p − r , q + r − .(This has just finitely many nonzero terms since ∂ rp , q = r ≥ p .) Also define,for k ∈ Z , D k = M p + q = k E p , q ,and define δ k : D k → D k − as the map which restricts to ∂ p , q on the respectivesummands E p , q . Each D k has a filtration given by F s D k = M p + q = k , p ≤ s E p , q ,which is consistent with (A.1) by (A.2). By definition, the map δ k respects thisfiltration, and indeed satisfies the stronger property δ k ( F s D k ) ⊂ F s − D k − .We will now compute the kernel and image of δ k . For a general element x = P p x p ∈ D k where each x p ∈ E p , k − p , the component of δ k x in the summand E m , k − − m ⊂ D k − is equal to X r ∂ rm + r , k − m − r x m + r . YMPLECTICALLY KNOTTED CODIMENSION-ZERO EMBEDDINGS OF DOMAINS IN R Now ∂ rm + r , k − m − r x m + r lies in the subspace B r + m , k − − m ∩ H rm , k − − m of E m , k − − m .But these latter subspaces are independent as r varies: indeed given finitelymany elements y r ∈ B r + m , k − − m ∩ H rm , k − − m that are not all zero, if r max is chosenmaximal subject to the property that y r max = = y r max ∈ H r max m , k − − m while for all s < r max we have y s ∈ B s + m , k − − m ⊂ B r max m , k − − m would implythat P y r = H r max m , k − − m is complementary to B r max m , k − − m .The independence of these subspaces implies that, for x p ∈ E p , k − p , the com-ponent of δ k €P p x p Š in E m , k − m − is zero only if each ∂ rm + r , k − m − r x m + r separatelyvanishes. Thus:(A.4) X p x p ∈ ker δ k ⇔ ( ∀ p , r )( ∂ rp , k − p x p = ) .Now fixing p and recalling that Z pp , k − p = Z ∞ p , k − p and ∂ rp , k − p = r ≥ p , notethat we have E p , k − p = Z ∞ p , k − p ⊕ M p − p , k − p ⊕ · · · ⊕ M p , k − p .Moreover, for r < p , ∂ rp , k − p vanishes on Z r + p , k − p ⊃ Z ∞ p , k − p and on each M jp , k − p for j = r while restricting injectively to M rp , k − p . Hence ∂ rp , k − p x p = all r if andonly if x p ∈ Z ∞ p , k − p . In combination with (A.4) this shows: Proposition A.5. ker ( δ k : D k → D k − ) = M p Z ∞ p , k − p and, for each s ∈ N , ker ( δ k | F s D k ) = M p ≤ s Z ∞ p , k − p .Next we will show: Proposition A.6. Im ( δ k : D k → D k − ) = M p B ∞ p , k − − p and, for s ∈ N , Im ( δ k | F s D k ) = M p < s B s − p + p , k − − p . Proof.
As noted earlier the summand of δ k €P p x p Š in E m , k − − m is P r ∂ rm + r , k − m − r x m + r ,which is a sum of terms in the mutually independent subspaces B r + m , k − − m ∩ H rm , k − − m . Note that, for fixed k , m and any t ∈ N ,(A.5) M ≤ r ≤ t € B r + m , k − − m ∩ H rm , k − − m Š = B t + m , k − − m :indeed using the inclusions B rm , k − − m ⊂ B r + m , k − − m ⊂ Z rm , k − − m = H rm , k − − m ⊕ B rm , k − − m we see that B r + m , k − − m = ( B r + m , k − − m ∩ H rm , k − − m ) ⊕ B rm , k − − m ; applying this induc-tively starting from B m , k − − m = { } yields (A.5). The same reasoning shows that L ∞ r = € B r + m , k − − m ∩ H rm , k − − m Š = B ∞ m , k − − m . Thus to prove the proposition it suf-fices to show that, given p ∈ N and elements y r ∈ B r + p − r , k + r − p − ∩ H rp − r , k + r − p − for 1 ≤ r < p , we can find a single x ∈ E p , k − p with ∂ rp , k − p x = y r for each r . But this is an easy consequence of Corollary A.4: using the decomposition E p , k − p = Z ∞ p , k − p ⊕ M p − p , k − p ⊕ · · ·⊕ M p , k − p we can take x to be an element with triv-ial component in Z ∞ p , k − p and with component in each respective M rp , k − p equal toa preimage of y r under ∂ rp , k − p . (cid:3) Corollary A.7.
Let D ∗ = ⊕ k D k and δ = ⊕ k δ k . Then ( D ∗ , δ ) is a filtered chaincomplex whose total homology is given by H k ( D ∗ , δ ) = ⊕ p Z ∞ p , k − p ⊕ p B ∞ p , k − p .Moreover, for s ∈ N , t ∈ N ∪ {∞} with s ≤ t we haveIm (cid:0) H k ( F s D ∗ , δ ) → H k ( F t D ∗ , δ ) (cid:1) = L p ≤ s Z ∞ p , k − p L p ≤ s B t − p + p , k − p . Proof.
That ( D ∗ , δ ) is a chain complex simply results from Propositions A.5 andA.6 and the fact that B ∞ p , q ⊂ Z ∞ p , q ; the computation of H k ( D ∗ , δ ) likewise followsimmediately. The computation of Im (cid:0) H k ( F s D ∗ , δ ) → H k ( F t D ∗ , δ ) (cid:1) also followsbecause this image is essentially by definition equal to the quotient of ker ( δ | F s D k ) by Im ( δ | F t D k + ) ∩ F s D k . (For the case that t = s , it perhaps also bears noting that B s , k − s = { } , so that ⊕ p < s B s − p + p , k − p = ⊕ p ≤ s B s − p + p , k − p ). (cid:3) Lemma A.1 now follows almost immediately from Corollary A.7 and Propo-sition A.3. Indeed, projecting away ˆ B p , k − p gives isomorphisms ˆ Z rp , k − p ˆ B rp , k − p ∼ = Z rp , k − p B rp , k − p so Corollary A.7 and Proposition A.3 show that we have, whenever s ∈ N and1 ≤ s ≤ t ≤ ∞ ,Im ( H k ( F s D ∗ , δ ) → H k ( F t D ∗ , δ )) ∼ = s M p = Im (cid:0) H k ( F p C ∗ , ∂ ) → H k ( F t C ∗ , ∂ ) (cid:1) Im (cid:0) H k ( F p − C ∗ , ∂ ) → H k ( F t C ∗ , ∂ ) (cid:1) .Since F C ∗ = { } , we can then iteratively choose complements to Im (cid:0) H k ( F p − C ∗ , ∂ ) → H k ( F t C ∗ , ∂ ) (cid:1) in Im (cid:0) H k ( F p C ∗ , ∂ ) → H k ( F t C ∗ , ∂ ) (cid:1) to obtain an isomorphism Im (cid:0) H k ( F s D ∗ , δ ) → H k ( F t D ∗ , δ ) (cid:1) ∼ = Im (cid:0) H k ( F s C ∗ , ∂ ) → H k ( F t C ∗ , ∂ ) (cid:1) . Moreover in the case that t = ∞ , as s varies this can be done in such a way that if s < s ′ then theisomorphism Im (cid:0) H k ( F s ′ D ∗ , δ ) → H k ( D ∗ , δ ) (cid:1) ∼ = Im (cid:0) H k ( F s ′ C ∗ , ∂ ) → H k ( C ∗ , ∂ ) (cid:1) restricts to Im (cid:0) H k ( F s D ∗ , δ ) → H k ( D ∗ , δ ) (cid:1) as the already-chosen isomorphism YMPLECTICALLY KNOTTED CODIMENSION-ZERO EMBEDDINGS OF DOMAINS IN R Im (cid:0) H k ( F s D ∗ , δ ) → H k ( D ∗ , δ ) (cid:1) ∼ = Im (cid:0) H k ( F s C ∗ , ∂ ) → H k ( C ∗ , ∂ ) (cid:1) ; hence by tak-ing the union over s we obtain an isomorphism H k ( D ∗ , δ ) ∼ = H k ( C ∗ , δ ) (corre-sponding to the case s = t = ∞ in Lemma A.1).Since we have already seen that our complex ( D ∗ , δ ) satisfies the other re-quired properties, this completes the proof of Lemma A.1.A.2. Construction of
C C ∗ ( X , λ ) . Since we assume that the Reeb flow on theboundary of ( X , λ ) is nondegenerate, the set of actions (equivalently, periods) ofthe Reeb orbits on ∂ X is discrete; of course every element of this set is positive,so let us denote by T < T < · · · < T r < · · · the numbers which arise as actionsof Reeb orbits on ∂ X . Also write T =
0. By [ GH17, Proposition 3.1 ] , the maps ı L , L : C H L ( X , λ ) → C H L ( X , λ ) give a directed system ( i.e. ı L , L ◦ ı L , L = ı L , L ), and ı L , L is an isomorphism if the interval ( L , L ] does not contain anyof the actions T i . So in particular if L ≤ L ′ with L ∈ [ T i , T i + ) , L ′ ∈ [ T j , T j + ) then there is a commutative diagram(A.6) C H L ( X , λ ) ı L , L ′ / / C H L ′ ( X , λ ) C H T i ( X , λ ) ı Ti , L ∼ = O O ı Ti , Tj / / C H T j ( X , λ ) ı Tj , L ′ ∼ = O O where both vertical arrows are isomorphisms. So to understand the maps ı L , L it suffices to understand the maps ı T i , T j .By definition ( [ GH17, Definition 6.1 ] ), we have C H L ( X , λ ) = lim −→ N , H H F S , N , + , ≤ L ( H , J ) where the direct limit is taken over parametrized Hamiltonians H : S × ˆ X × S N + → R on the Liouville completion ˆ X of X that satisfy a certain admissi-bility condition, with the structure maps being given by parametrized versionsof continuation maps associated to pairs ( N , H ) , ( N ′ , H ′ ) with N ≤ N ′ , H ≤ H ′ | S × ˆ X × S N + . Here H F S , N , + , ≤ L ( H , J ) is the homology of the subcomplex (whichfor brevity we will denote by C ( N , H ) L ) generated by orbits of symplectic actionat most L of the positive equivariant Floer complex C ( N , H ) ∞ : = C F S N , + ( N , H ) C F S N , + , ≤ ǫ ( N , H ) where 0 < ǫ ≪ T . The maps ı L , L : C H L ( X , λ ) → C H L ( X , λ ) are by defi-nition the maps induced on the direct limit by the maps H F S , N , + , ≤ L ( N , H ) → H F S , N , + , ≤ L ( N , H ) given by the inclusion of subcomplexes C ( N , H ) L , → C ( N , H ) L .Suppose that { ( N i , H i ) } ∞ i = is any cofinal, linearly ordered subset of the par-tially ordered set of pairs ( N , H ) used to define C H L ( X , λ ) . We can then formthe direct limit of the chain complexes C ( N i , H i ) ∞ , using as structure maps thecompositions of chain level continuation maps C ( N i , H i ) ∞ → C ( N i + , H i + ) ∞ . Denote this direct limit by C −→ . Since the continuation maps preserve the fil-tration by symplectic action, for any L ∈ R we likewise have a direct limit C −→ L = lim −→ i C ( N i , H i ) L , and the C −→ L form an R -valued filtration of C −→ .Let us coarsen this R -filtration to an N filtration by, for each p ∈ N , choosing T ′ p with T p < T ′ p < T p + , and letting F p C −→ = C −→ T ′ p (Recall our notation that T = T p for p > ∂ X , in increasing order.) As in [ GH17, Remark 5.6 ] , for i suf-ficiently large every generator of C ( N i , H i ) will have filtration level greater than T ′ , so that C ( N i , H i ) T ′ = { } for i sufficiently large and so F C −→ = { } . The factthat ∪ p C ( N i , H i ) T ′ p = C ( N i , H i ) for each i implies that likewise ∪ p F p C −→ = C −→ . Allof our complexes are Z -graded because of the assumption that c ( T X ) | π ( X ) = C −→ , producing a filtered com-plex ( D ∗ , δ ) with F r D ∗ = M ≤ p ≤ r H ∗ lim −→ i C ( N i , H i ) T ′ p lim −→ i C ( N i , H i ) T ′ p − such that for each k ∈ Z , s ≤ t we haveIm (cid:0) H k ( F s D ∗ , δ ) → H k ( F t D ∗ , δ ) (cid:1) ∼ = Im € H k ( C −→ s ) → H k ( C −→ t ) Š ;note that (for finite t ) the right-hand side is precisely the image of ı T s , T t ingrading k . Also, since lim −→ is an exact functor, we have H ∗ lim −→ i C ( N i , H i ) T ′ p lim −→ i C ( N i , H i ) T ′ p − ∼ = lim −→ i H ∗ ‚ C ( N i , H i ) T ′ p C ( N i , H i ) T ′ p − Œ .Thus we have a filtered complex ( D ∗ , δ ) whose r -filtered part is F r D ∗ = lim −→ i M ≤ p ≤ r H ∗ ‚ C ( N i , H i ) T ′ p C ( N i , H i ) T ′ p − Œ and such that, for 1 ≤ s ≤ t < ∞ ,(A.7)Im (cid:16) ı T ′ s , T ′ t : C H T ′ s k ( X , λ ) → C H T ′ t k ( X , λ ) (cid:17) ∼ = Im (cid:0) H k ( F s D ∗ , δ ) → H k ( F t D ∗ , δ ) (cid:1) .The foregoing discussion applies to an arbitrary cofinal linearly ordered sub-set { ( N i , H i ) } ∞ i = of the set of admissible pairs ( N , H ) . For a particular choice ofsuch a cofinal subset consisting of Hamiltonians as described in [ G15, Section3.1 ] and [ GH17, Remark 5.15 ] , the homologies H ∗ (cid:18) C ( N i , H i ) T ′ p C ( N i , H i ) T ′ p − (cid:19) are computed YMPLECTICALLY KNOTTED CODIMENSION-ZERO EMBEDDINGS OF DOMAINS IN R in [ G15, Section 3.2 ] , [ GH17, Section 6.7 ] . Namely, the space H ∗ (cid:18) C ( N i , H i ) T ′ p C ( N i , H i ) T ′ p − (cid:19) is generated by elements b γ and u N i ⊗ b γ as γ ranges over good Reeb orbits hav-ing action equal to T p ; writing C Z for the Conley–Zehnder index, the gradingof b γ is C Z ( γ ) and that of u N i ⊗ b γ is C Z ( γ ) + N i +
1. The continuation maps H ∗ (cid:18) C ( N i , H i ) T ′ p C ( N i , H i ) T ′ p − (cid:19) → H ∗ (cid:18) C ( N i + , H i + ) T ′ p C ( N i + , H i + ) T ′ p − (cid:19) moreover map b γ to b γ and u N i ⊗ b γ tozero, as one can see based on [ BO13a, Remark 3.7 ] . Thus in any given degree k the direct limit lim −→ i H k (cid:18) C ( N i , H i ) T ′ p C ( N i , H i ) T ′ p − (cid:19) has basis in bijection with the good Reeborbits on ∂ X of action T p and Conley-Zehnder index k .So the N -filtered complex ( D ∗ , δ ) produced by Lemma A.1 has the propertythat F r D k is the span of a set of generators in bijection with the good Reeborbits on ∂ X of Conley–Zehnder index k and action at most T r . The complex C C ∗ ( X , λ ) promised in Lemma 2.1 is then given by converting ( D ∗ , δ ) into an R -filtered complex by taking the L -filtered part C C L ∗ ( X , λ ) to be equal F r D ∗ where r is maximal subject to the condition that T r ≤ L . In particular we haveequalities C C L ∗ ( X , λ ) = C C L ′ ∗ ( X , λ ) whenever L , L ′ ∈ [ T r , T r + ) . Since δ strictlydecreases the N -filtration on D ∗ , it likewise strictly decreases this R -filtration.By (A.7), we have isomorphismsIm (cid:16) ı T ′ s , T ′ t : C H T ′ s k ( X , λ ) → C H T ′ t k ( X , λ ) (cid:17) ∼ = Im (cid:16) H k (cid:0) C C T ′ s ∗ ( X , λ ) (cid:1) → H k (cid:0) C C T ′ t ∗ ( X , λ ) (cid:1)(cid:17) for s ≤ t , and then by applying (A.6) we obtain a similar isomorphism with T ′ s , T ′ t replaced by arbitrary L , L ′ with L ≤ L ′ . The special case that L = L ′ shows that C H Lk ( X , λ ) is isomorphic to H k (cid:0) C C L ∗ ( X , λ ) (cid:1) since in this case therelevant inclusion-induced map is the identity. This completes the proof that thefiltered complex C C ∗ ( X , λ ) = ∪ L C C L ∗ ( X , λ ) with boundary operator δ satisfiesthe properties required by Lemma 2.1.R EFERENCES [ BCE07 ] F. Bourgeois, K. Cieliebak, and T. Ekholm.
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