Slice genus, T-genus and 4-dimensional clasp number
SSLICE GENUS, T –GENUS AND –DIMENSIONAL CLASP NUMBER DELPHINE MOUSSARD
Abstract.
The T –genus of a knot is the minimal number of borromean-type triple points ona normal singular disk with no clasp bounded by the knot; it is an upper bound for the slicegenus. Kawauchi, Shibuya and Suzuki characterized the slice knots by the vanishing of their T –genus. We generalize this to provide a –dimensional characterization of the slice genus.Further, we prove that the T –genus majors the –dimensional positive clasp number and wededuce that the difference between the T –genus and the slice genus can be arbitrarily large. Weintroduce the ribbon counterpart of the T –genus and prove that it is an upper bound for theribbon genus. Interpreting the T –genera in terms of ∆ –distance, we show that the T –genus andthe ribbon T –genus coincide for all knots if and only if all slice knots are ribbon. We work inthe more general setting of algebraically split links and we also discuss the case of colored links.Finally, we express Milnor’s triple linking number of an algebraically split –component link asthe algebraic intersection number of three immersed disks bounded by the three components. Contents
1. Introduction 12. Definitions and main statements 23. Surfaces, cobordisms and projections 64. T –genera and ∆ –distance 115. Arf invariant and Milnor’s triple linking number 136. Examples 157. Colored links 17References 191. Introduction
The first attempts to prove the ribbon–slice conjecture led to a 3–dimensional characterizationof slice knots. Kawauchi, Shibuya and Suzuki [KSS83] proved that slice knots are exactly thosethat bound a normal singular disk in the 3–sphere with no clasp and no triple point of a certaintype, called here borromean. We generalize this result to a 3–dimensional characterization ofthe slice genus, proving that the slice genus of a knot is the minimal genus of a normal singularsurface in the 3–sphere, with no clasp and no borromean triple point, bounded by the knot.It was proved by Kaplan [Kap79] that any knot bounds a normal singular disk with no clasp.This allowed Murakami and Sugishita [MS84] to define the T –genus of a knot as the minimalnumber of borromean triple points on such a disk. They proved that the T –genus is a concordanceinvariant and an upper bound for the slice genus. Further, they showed that the mod– reductionof the T –genus coincides with the Arf invariant. From these properties, they deduced the value a r X i v : . [ m a t h . G T ] J a n DELPHINE MOUSSARD of the T –genus for several knots for which the difference between the T –genus and the slice genusis 0 or 1. This raises the question of wether this difference can be greater than one; we provethat it can be arbitrarily large. For this, we show that the T –genus is an upper bound for the4–dimensional positive clasp number and we use a recent result of Daemi and Scaduto [DS20]that states that the difference between the 4–dimensional positive clasp number and the slicegenus can be arbitrarily large.We introduce the ribbon T –genus of a knot, defined as the minimal number of borromeantriple points on an immersed disk bounded by the knot with no clasp and no non-borromeantriple point. In [KMS83], Kawauchi, Murakami and Sugishita proved that the T –genus of aknot equals its ∆ –distance to the set of slice knots. We prove the ribbon counterpart of it,namely that the ribbon T –genus of a knot equals its ∆ –distance to the set of ribbon knots. Asa consequence, providing a knot with distinct T –genus and ribbon T –genus would imply theexistence of a non-ribbon slice knot.We generalize the definition and properties of the T –genus to algebraically split links. Inaddition, we express Milnor’s triple linking number of an algebraically split 3–component link asthe algebraic intersection of three disks bounded by the three components, that intersect onlyalong ribbons and borromean triple points. We also give an elementary proof, in the setting ofnon-split links, that the difference between the T –genus and the slice genus can be arbitrarilylarge, computing the T –genus on a family of cabled borromean links. Finally, we discuss thecase of colored links. Conventions.
We work in the smooth category. All manifolds are oriented. Boundaries oforiented manifolds are oriented using the “outward normal first” convention.
Acknowledgments.
I wish to thank Emmanuel Wagner for motivating conversations and JaeChoon Cha for an interesting suggestion.2.
Definitions and main statements ∂ Σ ib ribbon ∂ Σ clasp Figure 1.
Lines of double points and their preimagesIf Σ is a compact surface immersed in S , the self-intersections of Σ are lines of double points,which possibly intersect along triple points. The lines of double points are of two kinds: ribbonsand clasps (see Figure 1). A ribbon is a line of double points whose preimages by the immersionare a b –line properly immersed in Σ and an i –line immersed in the interior of Σ . A clasp isa line of double points that is not a ribbon. When three ribbons meet at a triple point, thereare again two possibilities. We say that the triple point is borromean if its three preimages are LICE GENUS, T –GENUS AND –DIMENSIONAL CLASP NUMBER 3 intersections of a b –line and an i –line (see Figure 2). We will also consider surfaces with branchpoints , namely points that have a neighborhood as represented in Figure 3. p p p borromean p p p non borromean Figure 2.
Triple points on a disk
The picture represents the singular set of the disk on its preimage.The points p , p and p are the three preimages of a triple point p . A compact surface is: • normal singular if it is immersed in S except at a finite number of branch points, • ribbon if it is immersed in S with no clasp and no triple point, • T –ribbon if it is immersed in S with no clasp and no non-borromean triple point, • slice if it is smoothly properly embedded in B . • Figure 3.
A branched pointBeyond ribbons and clasps, the different types of double point lines that appear on a normalsingular surface are closed lines, namely circles , or intervals, called branched ribbons if one end-point is branched and branched circles if the two endpoints are branched, see Figure 4, where thepreimages are drawn. Like for a ribbon, the two preimages of a branched ribbon are naturallydivided into a b –line that contains a boundary point and an i –line that doesn’t. For a (branched)circle, one may call b –line one preimage and i –line the other. A normal singular surface withsuch namings assigned to the preimages of each (branched) circle is said to be marked . A triplepoint on a normal singular surface is borromean if its three preimages are intersections of a b –lineand an i –line.Given a link L = L (cid:116) · · · (cid:116) L n in S , a complex for L is a union of compact surfaces Σ = ∪ ≤ i ≤ n Σ i such that ∂ Σ i = L i for all i . The genus of such a complex is the sum of the generaof its components. The slice genus g s ( L ) (resp. ribbon genus g r ( L ) ) of a link L is the minimalgenus of a slice (resp. ribbon) complex for L . Note that g s ( L ) ≤ g r ( L ) . These invariantsare well-defined (finite) for algebraically split links , namely links whose components have trivialpairwise linking numbers. The following result generalizes the characterization of slice knots byKawauchi–Shibuya–Suzuki [KSS83, Corollary 6.7]. DELPHINE MOUSSARD ∂ Σ ib ribbon ∂ Σ ib branched ribbon ∂ Σ circle ∂ Σ branched circle Figure 4.
Double points lines on a normal singular surface (preimages)
Theorem (Corollary 3.5) . The slice genus of an algebraically split link L equals the minimalgenus of a marked normal singular complex for L with no clasp and no borromean triple point. The T –genus T s ( L ) (resp. ribbon T –genus T r ( L ) ) is the minimal number of borromean triplepoints on a marked normal singular disks complex (resp. T –ribbon disks complex) with no claspfor L ; obviously T s ( L ) ≤ T r ( L ) . Note that these numbers may be undefined. Kaplan [Kap79]proved that any algebraically split link bounds a T –ribbon disks complex. In particular, the(ribbon) T –genus of an algebraically split link is well-defined. It follows that any of the fourinvariants g s , g r , T s , T r is well-defined on a link if and only if this link is algebraically split (seefor instance Corollary 4.3).In [MS84], Murakami and Sugishita proved that the T –genus is an upper bound for the slicegenus. We generalize this result to algebraically split links and prove the ribbon counterpart ofit. Theorem (Corollary 4.3) . For any algebraically split link L , g s ( L ) ≤ T s ( L ) and g r ( L ) ≤ T r ( L ) . The proof we give relies on the expression of the T –genus in terms of ∆ –distance, which is thedistance on the set of links defined by the ∆ –move (see Figure 5). The ∆ –slicing number s ∆ ( L ) Figure 5. ∆ –move(resp. ∆ –ribonning number r ∆ ( L ) ) is the minimal number of ∆ –moves necessary to change L into a slice (resp. ribbon) link. Note that these are well-defined if and only if L is algebraicallysplit (see Theorem 4.1). Kawauchi, Murakami and Sugishita [KMS83] proved that the T –genusof a knot equals its slicing number. We generalize this as follows. Theorem (Theorem 4.2) . For any algebraically split link L , T s ( L ) = s ∆ ( L ) and T r ( L ) = r ∆ ( L ) . It is a natural generalization of the ribbon–slice question to ask wether the T –genus and theribbon T –genus always coincide. The above result shows that it is an equivalent question. Corollary 2.1.
The slice knots are all ribbon if and only if T s ( K ) = T r ( K ) for any knot K .The same holds for algebraically split links with a given number of components. LICE GENUS, T –GENUS AND –DIMENSIONAL CLASP NUMBER 5 In [MS84], Murakami and Sugishita proved that the T –genus of knots is a concordance in-variant whose mod– reduction is the Arf invariant — we generalize this to algebraically splitlinks. They deduced the value of the T –genus for several knots satisfying T s − g s = 0 , . Thequestion arises then to know if this difference can be arbitrarily large. In Section 6, we providea family of non-split links B n with g s ( B n ) = 1 and T s ( B n ) = n . These links are constructed bycabling one component of the borromean link. Nevertheless, this is not fully satisfying in thatit is based on the augmentation of the number of components of the link. To get the answer inthe setting of knots, we will compare the T –genus with the –dimensional positive clasp number.The –dimensional clasp number c ( L ) of a link L is the smallest number of transverse doublepoints on an immersed disks complex in B bounded by L . Similarly define the –dimensionalpositive/negative clasp number c +4 ( L ) / c − ( L ) by counting the positive/negative double points.We also consider here a balanced version of this invariant, which is the most natural in thecomparison with the T –genus. The balanced –dimensional clasp number c b ( L ) of a link L is thesmallest number of positive (or negative) transverse double points on an immersed disks complexin B , bounded by L , with trivial self-intersection number. We have the following immediate in-equalities (note that a positive or negative transverse double point can be added to an immersedsurface in B without modifying its boundary). c +4 ( L ) , c − ( L ) ≤ c b ( L ) ≤ c ( L ) ≤ c b ( L ) Note that, among all the invariants introduced at that point, only c ± depends on the orienta-tion, in the case of a link with at least two components.It is well known that the –dimensional clasp number is an upper bound for the slice genus —a transverse double point can be smoothed at the cost of adding one to the genus of the surface.For knots, this can be improved as follows. Lemma (Lemma 3.3) . For any knot K , g s ( K ) ≤ c b ( K ) . We will see in Section 6 that this lemma does not hold for algebraically split links. In contrast,we have the following.
Theorem (Corollary 3.2) . For any algebraically split link L , c b ( L ) ≤ T s ( L ) . In [DS20], Daemi and Scaduto minored the difference c +4 ( K ) − g s ( K ) for the connected sumsof copies of the knot . Theorem 2.2 (Daemi–Scaduto) . For any positive integer n , c +4 ( (cid:93) n ) − g s ( (cid:93) n ) ≥ n/ . Corollary 2.3.
The difference between the T –genus and the slice genus, for knots in S , can bearbitrarily large. This raises the question of what can be the difference between the T –genus and the balanced4–dimensional clasp number. In [Mil20], Miller proves the existence of knots with arbitrarilylarge slice genus and trivial positive and negative 4–dimensional clasp number, which impliesthat the difference T s − c ± can be arbitrarily large. Question 2.4.
Can the difference T s − c b be arbitrarily large for knots? In Section 6, we propose a family of knots that could realize such an unbounded difference,namely the family of twist knots. The 4–dimensional clasp number equals for all non-slice twistknots; I would conjecture that their T –genus raises linearly with the number of twists. DELPHINE MOUSSARD Surfaces, cobordisms and projections
In this section, we present some manipulations on surfaces and cobordisms and we deduce acomparison of the T –genus with the slice genus and the balanced –dimensional clasp number,and a characterization of the slice genus.On the preimage of a marked normal singular compact surface, a preimage of a triple point isa triple point of type ( b – i ) if it is the intersection of a b –line and an i –line. Proposition 3.1.
Let Σ be a marked normal singular compact surface in S with b borromeantriple points and no clasp. Then Σ is the radial projection of a properly immersed surface in B with b positive and b negative double points.Proof. Let ˜Σ be a compact surface and let ι : ˜Σ → S be a map which is an immersion exceptat a finite number of branch points, such that ι ( ˜Σ) = Σ . We will define a radius fonction on ˜Σ in order to immerse it in B . Let r : ˜Σ → (0 , be a smooth function that sends: • ∂ ˜Σ onto , • branch points and triple points of type ( b – i ) onto , • other points in an i –line into (0 , ) , • other points in a b –line into ( , .Note that, within the three preimages of a non-borromean triple point, only one has type ( b – i ),so that only one is sent onto . The map from ˜Σ to B = S × [0 , x, ∼ ( y, given by x (cid:55)→ ( x, r ( x )) is anembedding except at borromean triple points which are still triple points. It remains to modifythe radius function around these borromean triple points. r r r r ˜ (cid:96) ij • ˜ (cid:96) ˜ (cid:96) ˜ (cid:96) Figure 6.
Modification of the radius function around a triple point
The line (cid:96) ij is the ribbon line on Σ containing p whose preimage ˜ (cid:96) ij is made of an i –line containing p i and a b –line containing p j . Take a smooth function f : D → [0 , ε ] , where ε > , such that f = 0 on a collar neighborhoodof S , f (0) = ε and is the only critical point of f . Fix a borromean triple point p on Σ and write p , p , p its preimages, chosing the indices so that the i –line containing p has the same image asthe b –line containing p . Now consider small disk neighborhoods of p and p and define a newradius function r (cid:48) by r (cid:48) = r − f in the neighborhood of p , r (cid:48) = r + f in the neighborhood of p and r (cid:48) = r elsewhere. This desingularizes the triple point but adds two double points of oppositesigns with preimages on the i –line containing p and on the b –line containing p , see Figure 6. Ifthe neighborhoods and ε are small enough, no other singularity is created. Performing a similarmodification around each borromean triple point, we finally get the required immersion. (cid:3) LICE GENUS, T –GENUS AND –DIMENSIONAL CLASP NUMBER 7 Corollary 3.2.
For any algebraically split link L , c b ( L ) ≤ T s ( L ) . For knots, this corollary and the following lemma give a relation between the slice genus andthe T –genus. Neverthless, the lemma does not hold for algebraically split links and we will givealternative proofs of this relation. Lemma 3.3.
For any knot K , g s ( K ) ≤ c b ( K ) .Proof. Take a disk Σ properly immersed in B with ∂ Σ = K that realizes c b ( K ) . We willdesingularize Σ by tubing once for each pair of double points with opposite signs. Fix a pair ( p + , p − ) of respectively a positive and a negative self-intersection point of Σ . Take a path γ on Σ joining p + and p − , whose interior does not meet the singularities of Σ . Inside a neighborhoodof γ in B , one can find a solid tube C = γ × D where γ × { } = γ , such that C meets theleaf of Σ transverse to γ around p + (resp. p − ) along p + × D (resp. p − × D ), and C meets theleaf of Σ containing γ along γ . Now remove from Σ the interior of the disks p ± × D and reglue γ × S instead. The sign condition ensures that the new surface is again oriented. (cid:3) Proposition 3.1 says in particular that the boundary of a marked normal singular genus– g complex bounds a genus– g slice complex. We now prove the converse in order to get a three-dimensional characterization of the slice genus. Theorem 3.4.
Let S be a properly embedded compact surface in B . Then S is isotopic, viaan isotopy of properly embedded surfaces, to a surface whose radial projection in S is a markednormal singular compact surface with no clasp and no borromean triple point.Proof. Up to isotopy, we can assume that the radius function on S is a Morse function whoseindex (cid:96) critical points take the value ε (cid:96) with < ε < ε < ε < . Hence S can be reorganizedas a PL surface with: • at r = ε , a family ∆ of disjoint disks, • at r ∈ ( ε , ε ) , the boundary ∂ ∆ , • at r = ε , ∂ ∆ ∪ B , where B a disjoint union of embedded bands [0 , × [0 , that meet ∂ ∆ exactly along [0 , × { , } , • at r ∈ ( ε , ε ) , the split union of L = ∂S with a trivial link, • at r = ε , the disjoint union of L with a disjoint union D of disks bounded by the abovetrivial link, • at r ∈ ( ε , , the link L .Projecting radially S on S , we obtain a complex Σ = Σ (cid:48) ∪ D , where Σ (cid:48) is a ribbon complex —projection of ∆ ∪ B — and D is a family of disjoint disks, disjoint from L , that we can assumetransverse to Σ (cid:48) . Denote ˜Σ = ˜Σ (cid:48) ∪ ˜ D the preimage of Σ with the corresponding decomposition.Note that a double point p ∈ ∂D is a double point of Σ (cid:48) , all branch points of Σ lie on ∂D ,branch points are simple. We now analyse the lines of double points on Σ and their preimages.Let γ ⊂ Σ be the closure of a line of double points. Note that it cannot have an endpoint in Int( D ) since D is disjoint from L .Case 1: γ does not contain any branch point. Let ˜ γ be one of its two preimages. If ˜ γ ⊂ ˜ D , then γ is a circle and we mark ˜ γ as a b –line. If ˜ γ meets both ˜ D and ˜Σ (cid:48) , then it contains an intervalproperly embedded in ˜ D . It has to be continued on both sides by b –lines of ˜Σ (cid:48) . These end eitheron ∂ ˜Σ or on ∂ ˜ D . In the latter case, they are continued by an interval properly embedded in ˜ D .Iterating, we see that ˜ γ is either an interval with two endpoints on ∂ ˜Σ or a circle. In both cases, DELPHINE MOUSSARD we mark it as a b –line. Now assume ˜ γ ⊂ ˜Σ (cid:48) . If it is an interval, it has no endpoint on ∂ ˜ D , sothat it is a b –line or an i –line of ˜Σ (cid:48) . If it is a circle, either the other preimage of γ meets ˜ D , inwhich case we mark ˜ γ as an i –line, or the two preimages of γ are contained in ˜Σ (cid:48) and we assignthem different markings.Case 2: γ contains a single branch point p . Let ˜ γ be the whole preimage of γ and let ˜ p be thepreimage of p . Near p , γ lies in Σ (cid:48) ∩ D . Let q be the first point of ∂D reached from p . Thepreimages of q are ˜ q ∈ ∂ ˜ D and ˜ q ∈ Int( ˜Σ (cid:48) ) , see Figure 7. The curve ˜ γ joins ˜ p to ˜ q inside ˜ D and to ˜ q inside ˜Σ (cid:48) . It is continued from ˜ q by a b –line in ˜Σ (cid:48) and from ˜ q by an i –line in Σ (cid:48) . The b –line may end on ∂ ˜ D , in which case we iterate the argument, or on ∂ ˜Σ . Finally, the preimageof γ containing ˜ q is a b –line ending on ∂ ˜Σ and the preimage containing ˜ q is an i –line containedin ˜Σ (cid:48) . ∂ ˜Σ˜ D • ˜ p • ˜ q • ˜ q • • • • • • Figure 7.
Case of a single branch point p Case 3: γ contains two branch points, ie it is an interval with branch points as endpoints. Thetwo preimages of γ are intervals in ˜Σ with the same endpoints. Analyzing the situation from abranch point as in the previous case, we see that one preimage of γ is contained in ˜Σ (cid:48) , while theother meets ˜ D . We mark the first one as an i –line and the second one as a b –line.Finally Σ is a marked normal singular complex. Let p be a triple point of Σ . Since Σ (cid:48) is aribbon complex, p must have at least one preimage ˜ p in Int( ˜ D ) . It follows from the previousdiscussion that no i –line meets Int( ˜ D ) , so that ˜ p is the intersection of two b –lines. Hence p isnon-borromean. (cid:3) Corollary 3.5.
The slice genus of an algebraically split link L equals the minimal genus of amarked normal singular complex for L with no clasp and no borromean triple point. A cobordism from a link L to a link L (cid:48) is a surface S properly embedded in S × [0 , , suchthat ∂S = L (cid:48) × { } − L × { } ; the links L and L (cid:48) are said to be cobordant . The cobordism S is strict if it has genus and distinct components of L belong to distinct components of S ;note that the relation induced on links is not symmetric. A cobordism is a concordance if S isa disjoint union of annuli and each annulus has one boundary component in S × { } and theother in S × { } ; the links L and L (cid:48) are then said to be concordant . Theorem 3.6.
Let Σ be a marked normal singular compact surface in S with b borromeantriple points and no clasp. Let S be a compact surface properly embedded in S × [0 , such that ∂S ∩ (cid:0) S × { } (cid:1) = − ∂ Σ × { } . Then, up to an isotopy of S fixing the boundary, the image of S ∪ (Σ × { } ) by the projection S × [0 , (cid:16) S is again a marked normal singular compactsurface with b borromean triple points and no clasp. LICE GENUS, T –GENUS AND –DIMENSIONAL CLASP NUMBER 9 Proof.
The proof is essentially the same as for Theorem 3.4. This proof still works if the surface S ,at r = ε , is not a disjoint union of disks but a marked normal singular complex S . One can askthat, in the projected complex Σ , the bands added at r = ε avoid the singularities of S and theunion of disks D avoids the triple points of S . The same discussion shows that Σ can be markedas a normal singular surface with as many borromean triple points as S . Here, we consider S in S × [0 , with < ε < ε < ε = 1 and we project on S × { } . (cid:3) Corollary 3.7.
The T –genus of algebraically split links is a concordance invariant. ∼ Figure 8.
The borromean link and a T –ribbon disks complex for itThe next result says that a marked normal singular surface in S can be pushed into S × [0 , in order to isolate the borromean triple points. Proposition 3.8.
Let Σ be a marked normal singular surface in S with b borromean triplepoints and no clasp. Then there is a surface S properly embedded in S × [0 , and a disjointunion ∆ of b T –ribbon disks complexes as represented in Figure 8 such that S ∩ ( S ×{ } ) = − ∂ ∆ , S ∩ ( S × { } ) = ∂ Σ and the image of S ∪ ∆ by the projection S × [0 , (cid:16) S is Σ .Proof. Let ι : ˜Σ → S be an immersion except at a finite number of branch points, with image Σ .We will define a height fonction on ˜Σ in order to immerse it in S × [0 , as a cobordism from ∂ Σ to a split union of borromean links.First define a subsurface ˜ C of ˜Σ as the disjoint union of: • a small disk around each triple point of type ( b – b ), namely intersection of two b –lines, • for each b –line, a small disk around a point of the line between any two consecutive triplepoints of type ( b – i ), • for each closed b –line with no triple point, a small disk around any point of the circle, • at each branched point, a small disk meeting the b –line along an open interval admittingthe branched as an endpoint, • a collar neighborhood of ∂ Σ .Set ˜Σ = ˜Σ \ Int( ˜ C ) , see Figure 9. In restriction to ˜Σ , ι is an immersion whose image is a T –ribbon genus– surface such that any ribbon line contains at most one triple point. Now define ˜Σ as a neighborhood in ˜Σ of all the i –lines. Set ˜ C = ˜Σ \ Int( ˜Σ ) . Then ˜Σ is a dijoint unionof disks. On some of them, the restriction of ι to ˜Σ has no singularity; define ˜ C as their unionwith a collar neighborhood of the others. Set ˜Σ = ˜Σ \ Int( ˜ C ) . The restriction of ι to ˜Σ has,on each disk, one b –line and one i –line intersecting once. ˜Σ ˜ C ˜Σ ˜ C ˜Σ ˜ C Figure 9.
Pushing a normal singular surfaceLet h : ˜Σ → [0 , be a smooth function that sends: • ∂ ˜Σ onto , ∂ ˜Σ onto and ∂ ˜Σ onto , • Int( ˜ C ) into (0 , ) , Int( ˜ C ) into ( , ) and Int( ˜ C ) into ( , , • ˜Σ onto .Finally define an immersion ι (cid:48) : ˜Σ → S × [0 , by ι (cid:48) ( p ) = (cid:0) ι ( p ) , h ( p ) (cid:1) . Set S = ι (cid:48) (cid:16) ˜Σ \ Int( ˜Σ ) (cid:17) and ∆ = ι (cid:48) ( ˜Σ ) . (cid:3) The above results provide a characterization of the T –genus which generalizes a result ofKawauchi–Murakami–Sugishita in the case of knots [KMS83]. Corollary 3.9.
The T –genus of an algebraically split link L is the smallest integer b such thatthere is a strict cobordism from L to a split union of b borromean links.Proof. If Σ is a marked normal singular disks complex for L with b borromean triple points andno clasp, then Proposition 3.8 gives a strict cobordism from L to a split union of b borromeanlinks. Reciprocally, given such a cobordism S , define Σ as the disjoint union of disks boundedby the b borromean links as in Figure 8 and apply Theorem 3.6 to get a marked normal singulardisks complex for L with b borromean triple points and no clasp. (cid:3) Figure 10.
Ribbon complex for the borromean link
Corollary 3.10.
For any algebraically split link L , g s ( L ) ≤ T s ( L ) . LICE GENUS, T –GENUS AND –DIMENSIONAL CLASP NUMBER 11 Proof.
Figure 10 shows that the borromean link bounds a slice complex of genus . Take a strictcobordism from L to a split union of b = T s ( L ) borromean links and complete it into a slicecomplex of genus b for L by gluing a genus– slice complex to each borromean link. (cid:3) T –genera and ∆ –distance The ∆ –move on links represented in Figure 5 was introduced by Murakami and Nakanishi[MN89]. Note that it preserves the linking numbers between the components of the link. Thefollowing result gives the converse [MN89, Theorem 1.1]. Theorem 4.1 (Murakami–Nakanishi) . Two links are related by a sequence of ∆ –moves if andonly if they have the same number of components and their components have the same pairwiselinking numbers. A ∆ –move can be realized by gluing a borromean link , see Figure 11. This will be useful forrelating the T –genera and the ∆ –distance. The equality T s ( K ) = s ∆ ( K ) for a knot K was givenin [KMS83, Theorem 2].Borromean link Gluing a borromean link Figure 11.
Borromean link and ∆ –move Theorem 4.2.
For any algebraically split link L , T s ( L ) = s ∆ ( L ) and T r ( L ) = r ∆ ( L ) .Proof. The link L can be obtained from a slice link, which bounds a marked normal singulardisks complex Σ with no borromean triple point and no clasp, by s ∆ ( L ) ∆ –moves. Realize each ofthese ∆ –moves by gluing a borromean link. The borromean link bounds a complex of three disksthat intersect along three ribbons, with a single borromean triple point, see Figure 8. Hence,when gluing a borromean link, we glue to Σ such a complex of three disks with three bands. Thebands may meet Σ and create new ribbons, but we can assume that no other kind of singularityis added. Thus we obtain an immersed disks complex for L with s ∆ ( L ) borromean triple pointsand no clasp, proving T s ( L ) ≤ s ∆ ( L ) . Similarly, L can be obtained from a ribbon link, whichbounds a ribbon disks complex, by r ∆ ( L ) ∆ –moves, so that T r ( L ) ≤ r ∆ ( L ) . It remains to provethe reverse inequalities.We shall prove that T s ( L ) can be reduced by gluing a borromean link. There is a strictcobordism S from L to the split union of t = T s ( L ) borromean links, which we denote B t . This ∅ birthdeath fusionfission Figure 12.
Birth, death, fusion and fission movescobordism can be rearrange into the following pattern, where O k is a trivial k –component link, J is a link and the different steps are described in Figure 12. L L (cid:116) O k J B t (cid:116) O (cid:96) B t births fusions fissions deaths We can assume that each connected component of S contains one component of L and onecomponent of J . One can get a trivial 3–component link as a band sum of two borromean links,see Figure 13. Perform such a band sum gluing a borromean link B to our link B t . The bandsgluing B to B t can be glued before the fissions and deaths, hence be glued to J . Then thesebands can be slid to be glued onto parts of the link L . Hence we can start by gluing B to L and then perform the remaining of the cobordism. This provides a strict cobordism from L(cid:93)B to B t − (cid:116) O . Filling in the components of O with disks, we finally get a strict cobordism fromthe connected sum L(cid:93)B to B t − . In other words, we get a cobordism from a link L (cid:48) , obtainedfrom L by gluing a borromean link, to the disjoint union of t − borromean links; in particular T s ( L (cid:48) ) ≤ T s ( L ) − . It follows that s ∆ ( L ) ≤ T s ( L ) . Figure 13.
A trivial –component link as a band sum of two borromean linksNow, take a T –ribbon disks complex Σ for L with b = T r ( L ) borromean triple points. Let ˜Σ be its preimage. Assume there is a ribbon γ on Σ which contains more than one triple point;denote ˜ γ the corresponding b –line on ˜Σ . Take a path ˜ η on ˜Σ joining a point of ∂ ˜Σ to a pointof ˜ γ between two preimages of triple points, such that ˜ η does not meet any i –line — which ispossible since the i –lines are disjoint; denote η ⊂ Σ the image of ˜ η . Slide the boundary of Σ along η , see Figure 14. This results in an isotopy of L . Since ˜ γ may intersect some b –lines, someribbons may have been divided into more ribbons, but no other singularity appears. Moreover, γ has been divided into ribbons with less triple points. Finally, we can assume that any ribbonon Σ contains at most one triple point. LICE GENUS, T –GENUS AND –DIMENSIONAL CLASP NUMBER 13 ∂ ˜Σ˜ γ ˜ η (cid:32) ∂ ˜Σ Figure 14.
Sliding ∂ Σ along a pathFix a triple point p of Σ . For each preimage ˜ p of p , take a neighborhood in ˜Σ of the union ofthe corresponding i –line with a part of the corresponding b –line joining ˜ p to ∂ ˜Σ , see Figure 15.These neighborhoods are three disks with only three ribbons meeting at a borromean triple point.Cutting these three disks amounts to cutting a borromean link. The same modification can beperformed by a single ∆ –move. This implies r ∆ ( L ) ≤ T r ( L ) . (cid:3) ∂ ˜Σ ∼ ∂ ˜Σ Figure 15.
Isolate a borromean triple pointWe recover the result of Corollary 3.10 and get the ribbon counterpart of it.
Corollary 4.3.
For any algebraically split link L , g s ( L ) ≤ T s ( L ) and g r ( L ) ≤ T r ( L ) .Proof. By Theorem 4.2, T r ( L ) = r ∆ ( L ) , so that L can be obtained from a ribbon link, boundinga ribbon disks complex, by a sequence of T r ( L ) ∆ –moves. We have seen that a ∆ –move canbe realized by gluing a borromean link. This can be achieved by gluing to the ribbon complexrepresented in Figure 10, which has genus . The same holds for the slice version. (cid:3) Arf invariant and Milnor’s triple linking number
Recall that the Arf invariant is a Z / Z –valued concordance invariant of knots which takesthe value on the trivial knot and the value on the trefoil knot. The following result [Rob65,Theorem 2] allows to extend the Arf invariant to algebraically split links (Robertello gives thisresult more generally for the so-called proper links). Theorem 5.1 (Robertello) . Let L be an algebraically split link. If there are strict cobordismsfrom two knots K and K (cid:48) to L , then the Arf invariants Arf( K ) and Arf( K (cid:48) ) are equal. For an algebraically split link L , define the Arf invariant of L as Arf( L ) = Arf( K ) for anyknot K such that there is a strict cobordism from K to L . The following result was establishedin [KMS83] in the case of knots. Proposition 5.2.
Let L be an algebraically split link. Let Σ be a marked normal singular diskscomplex for L with no clasp and b borromean triple points. Then Arf( L ) = b mod .Proof. Let K be a knot obtained from L by merging the component, so that there is a strictcobordism from K to L . By Proposition 3.8, there is a strict cobordism from L to a split union B b of b borromean links. Composing these cobordisms gives a strict cobordism from K to B b .Hence Arf( K ) = Arf( L ) = Arf( B b ) . Now, the trefoil knot can be obtained from the borromeanlink by merging the three components, see Figure 16, hence Arf( B ) = 1 . This concludes sincethe Arf invariant is additive under connected sum and thus under split union. (cid:3) ∼ Figure 16.
A trefoil knot as a band sum of a borromean linkThis result provides an interesting geometric realization of the Arf invariant of an algebraicallysplit link L . The Arf invariant of L is not only the parity of T s ( L ) , but more generally the parityof the number of borromean triple points on any marked normal singular disks complex for L .This was first noticed by Kaplan for knots and T –ribbon disks [Kap79, Theorem 5.3].We now give an expression of Milnor’s triple linking number in terms of borromean triplepoints. The Milnor invariant µ ( L ) of an ordered –component link L = L (cid:116) L (cid:116) L is a Z –valuedconcordance invariant introduced by Milnor; we refer the reader to [Mil54] (see also [Mei18]) forits definition and main properties. Given a marked singular disks complex D = D ∪ D ∪ D for L with no clasp, define the borromean triple point number n bt ( D ) of D as the number ofborromean triple points of D involving its three components, counted with signs. Note thatchanging the orientation of one component of L , as well as permuting two components, changesthe sign of n bt ( D ) ; this also holds for µ ( L ) . Proposition 5.3.
Let L = L (cid:116) L (cid:116) L be an ordered algebraically split 3–component link. Let D = D ∪ D ∪ D be a marked singular disks complex for L . Then µ ( L ) = n bt ( D ) .Proof. Proposition 3.8 provides a strict cobordism S ⊂ S × [0 , from L ⊂ S × { } to a splitunion B t of t = T s ( L ) borromean links bounding a T –ribbon complex R t ⊂ S × { } , such that S ∪ R t projects onto D . This cobordism can be worked out as in the proof of Theorem 4.2 inorder to get a strict cobordism, from a connected sum L(cid:93)B to a sublink B t − of B t made of LICE GENUS, T –GENUS AND –DIMENSIONAL CLASP NUMBER 15 t − borromean links, which projects to a marked singular disks complex for L(cid:93)B . Note thatthe sign of the remaining borromean triple points is unchanged. Note also that the connectedcomponents of L involved in the connected sum correspond to the connected components of D involved in the cancelled triple point.If the borromean link is glued along one or two components of the initial link, then theinvariance of µ under link-homotopy — a relation which allows isotopy and self-crossing changeof each component — shows that the value of µ ( L ) is unchanged; as is the value of n bt ( D ) . If thethree components of the borromean link are glued to the three components of the initial link,then it is a result of Krushkal [Kru98, Lemme 9] that Milnor’s triple linking number is additiveunder such a gluing. Hence ± is added to µ ( L ) , depending on the orientations; the same valueis added to n bt ( D ) .Repeat this operation until there is no more borromean triple point on D . At that stage, n bt ( D ) = 0 and L has been turned to a slice link, so that µ ( L ) = 0 since µ is a concordanceinvariant. It follows that the initial values of n bt ( D ) and µ ( L ) were equal. (cid:3) Corollary 5.4.
Let L = L (cid:116) L (cid:116) L be an ordered algebraically split 3–component link. Then | µ ( L ) | ≤ T s ( L ) . Corollary 5.5.
Let L = L (cid:116) L (cid:116) L be an ordered algebraically split 3–component link. Let D = D ∪ D ∪ D be a T –ribbon disks complex for L . Then µ ( L ) equals the algebraic intersectionnumber (cid:104) D , D , D (cid:105) . Examples
Let B n be the link obtained by 0–cabling ( n − times a component of the borromean link B ,see Figure 17. Figure 17.
The link B Proposition 6.1.
For all n > , T r ( B n ) = T s ( B n ) = n and g r ( B n ) = g s ( B n ) = 1 .Proof. First note that the borromean link B satisfies T r ( B ) = T s ( B ) = 1 : Figure 8 gives a T –ribbon disks complex for B with one borromean triple point, and B is not slice since | µ ( B ) | = 1 .In Figure 10, taking n parallel copies of the green disk provides a genus–1 ribbon complex for B n ,so that g r ( B n ) ≤ . Moreover, B n is not slice since it has a non-slice sublink, namely B , thus g s ( B n ) ≥ . In Figure 17, the obvious disks bounded by the components of B n define a T –ribbondisks complex with exactly n borromean triple points, giving T r ( B n ) ≤ n . Now, take a markednormal singular disks complex D = D ∪ D (cid:48) ∪ ( ∪ ≤ i ≤ n D i ) for B n , where the disks D , . . . , D n arebounded by the parallel components of B n . For any ≤ i ≤ n , the number of borromean triplepoints on D defined by the intersection D ∩ D (cid:48) ∩ D i is at least 1 since B is not slice. Hence T s ( B n ) ≥ n . (cid:3) Let K n be the twist knot defined by n half-twists, see the left-hand side of Figure 18. It is easyto see that the genus of K n is . On the other hand, one can check that K and K are slice. Afamous result of Casson and Gordon [CG86] states that these are the only slice twist knots. n h a l f - t w i s t s ∆ –move = Figure 18.
A twist knot and a ∆ –move on it Theorem 6.2 (Casson–Gordon) . For n (cid:54) = 0 , , g s ( K n ) = 1 . Lemma 6.3.
For n (cid:54) = 0 , , c b ( K n ) = c ( K n ) = 1 .Proof. It is a simple and well-known fact that the unknotting number majors the 4–dimensionalclasp number. Clearly, the unknotting number of non-trivial twist knots equals . (cid:3) Proposition 6.4.
For all n ≥ , | T r ( K n +2 ) − T r ( K n ) | = | T s ( K n +2 ) − T s ( K n ) | = 1 .Proof. For n ≥ , Figure 18 shows that K n +2 can be turned into K n by a single ∆ –move, whichmodifies T s and T r by at most one according to Theorem 4.2. Proposition 5.2 shows that theyhave to be modified. (cid:3) Corollary 6.5.
For all n > , T r ( K n − ) ≤ n and T r ( K n ) ≤ n − .Proof. The move on Figure 18 changes K into the trivial knot, so that T r ( K ) = 1 . The knot K is slice. (cid:3) The following result shows the failure of Lemma 3.3 for algebraically split links.
Lemma 6.6.
Let L be the split union of a non-slice twist knot with its mirror image. Then g s ( L ) = 2 and c b ( L ) = 1 .Proof. The slice genus follows from that of non-slice twist knots. Since L is not slice, c b ( L ) > .Now, each component of L bounds a disk in B with exactly one double point and the twodisks can be chosen to be the mirror image of each other, in order to get two double points withopposite sign. (cid:3) This lemma implies that the difference g s − c b , and thus g s − T s , can be arbitrarily large forsplit links: take the split union of arbitrarily many copies of the link L in the lemma. LICE GENUS, T –GENUS AND –DIMENSIONAL CLASP NUMBER 17 Colored links A colored link is a link L in S with a given partition into sublinks L = L (cid:116) · · · (cid:116) L µ . A colored complex for L is a union of compact surfaces Σ = ∪ ≤ i ≤ µ Σ i such that ∂ Σ i = L i for all i .We have as previously notions of normal singular, ribbon, T –ribbon, slice complex and a notionof marking.Given two links K = (cid:116) ki =1 K i and J = (cid:116) (cid:96)j =1 J j , where the K i and J j are knots, the linkingnumber of K and J is lk ( K, J ) = (cid:80) ki =1 (cid:80) (cid:96)j =1 lk ( K i , J j ) . A colored link L is algebraically c –split if its sublinks have trivial pairwise linking numbers. We will generalize Kaplan’s result,proving that any algebraically c –split colored link bounds a T –ribbon genus– colored complex.Although Kaplan’s proof generalizes to this setting, we present here an alternative proof basedon Theorem 4.1. We start with preliminary results. Lemma 7.1.
Fix a positive integer n and integers (cid:96) ij for ≤ i < j ≤ n . There exists a link K with n connected components K i such that lk ( K i , K j ) = (cid:96) ij for all i < j and K bounds a genus– compact surface embedded in S .Proof. Take a trivial link K with n components K i . It bounds an embedded genus– surface Σ ,compact and connected. For given i < j , the linking lk ( K i , K j ) can be modified as follows.Take a band B = [0 , × [0 , on Σ such that { } × [0 ,
1] = B ∩ K i , { } × [0 ,
1] = B ∩ K j and (0 , × [0 , ⊂ Int(Σ) . Twist the band B around (cid:8) (cid:9) × [0 , , see Figure 19. This adds the numberof twists to the linking number lk ( K i , K j ) without modifying the other linking numbers. (cid:3) Σ K i K j Figure 19.
Twisting a band in Σ Lemma 7.2.
Let L = (cid:116) ≤ i ≤ µ L i be an algebraically c –split colored link. There exists an alge-braically c –split colored link J = (cid:116) ≤ i ≤ µ J i whose knot components have the same pairwise linkingnumbers as those of L and which bounds a T –ribbon genus– colored complex in S .Proof. First, thanks to Lemma 7.1, define J as the split union of links J i that realize the pairwiselinking numbers of the L i and bound embedded genus– surfaces Σ i in S . Write J i as the disjointunion of knots J i = (cid:116) ≤ (cid:96) ≤ k i J i(cid:96) . Fix i, j, (cid:96), m such that ≤ i < j ≤ µ , ≤ (cid:96) ≤ k i and ≤ m < k j .Take a band in Σ j joining J jm to J jk j . Link this band around J i(cid:96) in order to realize the desiredlinking lk ( J i(cid:96) , J jm ) , see Figure 20. This adds ribbon intersections on the complex ∪ µi =1 Σ i . Then,for (cid:96) < k i , realize the linking lk ( J i(cid:96) , J jk j ) using a band on Σ i joining J i(cid:96) to J ik i . Since J remainsalgebraically c –split, all linking numbers are finally realized. (cid:3) Proposition 7.3.
Any algebraically c –split colored link bounds a T –ribbon genus– coloredcomplex. Σ i J i(cid:96) Σ j J jm J jk j Figure 20.
Linking a band in Σ j around a component of J i Proof.
Let L be an algebraically c –split colored link. Lemma 7.2 provides another algebraically c –split colored link J with the same pairwise linking numbers, that bounds a T –ribbon genus– colored complex Σ . Theorem 4.1 says that L can be obtained from J by a sequence of ∆ –moves.Realize these ∆ –moves by gluing borromean links to L and associated T –ribbon disks complexesto Σ . This provides a ribbon disks complex for L . (cid:3) We consider a colored version of the invariants studied above, namely we define these invariantsfrom colored complexes, and we add a superscript c to distinguish them from the non-coloredversion. It follows from Proposition 7.3 that the slice and ribbon genera and the T –genera of analgebraically c –split colored link are well-defined. Most of the results we have seen remain truein the colored setting, with the same proof. We collect them in the next statement.A colored cobordism from a link L to a link L (cid:48) is a disjoint union in S × [0 , of genus– cobordisms from the colored sublinks of L to the colored sublinks of L (cid:48) . In this definition, somesublinks of the colored links may be empty. Theorem 7.4.
For any algebraically c –split colored link L , • the colored slice genus of L equals the minimal genus of a marked normal singular coloredcomplex for L with no clasp intersection and no borromean triple point, • the T –genus of L is the smallest integer b such that there is a colored cobordism from L to a split union of b borromean links with any coloring, • T cs ( L ) = s c ∆ ( L ) and T cr ( L ) = r c ∆ ( L ) , • g cs ( L ) ≤ T cs ( L ) and g cr ( L ) ≤ T cr ( L ) , • c b,c ( L ) ≤ T cs ( L ) .Proof. The first point is a corollary of Theorem 3.4. The second point follows from Theorem 3.6and Proposition 3.8. The proof of Theorem 4.2 works in the colored setting and gives the thirdpoint; the fourth is a corollary of it. Finally the fifth point is a corollary of Proposition 3.1. (cid:3)
Let L and L (cid:48) be colored links. A colored concordance from L to L (cid:48) is a disjoint union ofconcordances between the sublinks of L and L (cid:48) . Note that the relations in the next result arealso satisfied by the slice genus. Theorem 7.5.
Let L and J be algebraically c –split colored links. Let (cid:98) J be the colored link withthe same underlying link as J and a different color for each knot component. • If L and J are related by a colored cobordism, then T s ( L ) ≤ T s ( (cid:98) J ) . • If L and J are related by a colored concordance, then T s ( L ) = T s ( J ) . LICE GENUS, T –GENUS AND –DIMENSIONAL CLASP NUMBER 19 Proof.
First assume L and J are cobordant and define Σ as the union of a cobordism from L to J with a marked normal singular disks complex S for (cid:98) J that realizes T s ( (cid:98) J ) . Since S is madeof disks, after removing closed components if necessary, Σ is a marked normal singular genus– complex for L with at most T s ( (cid:98) J ) borromean triple points.Now assume L and J are concordant and do the same with a concordance from L to J and amarked normal singular genus– complex S for J that realizes T s ( J ) . Once again, Σ has genus ,so that T s ( L ) ≤ T s ( J ) . Similarly T s ( J ) ≤ T s ( L ) . (cid:3) ∆ Figure 21. A ∆ –move on a Hopf linkWe end with a point which does not generalize to colored links. In the non-colored setting,the parity of the number of borromean triple points on a marked normal singular complex isfixed, it only depends on the link. A consequence of this fact is that a ∆ –move performed onan algebraically split link always modifies the link. Figure 21 shows a ∆ –move on the Hopf linkthat leaves it unchanged. It follows that a colored Hopf link with a single color bounds T –ribbongenus– complexes with any number of borromean triple points. References [CG86]
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