SSOME COMPUTATIONS ON INSTANTON KNOT HOMOLOGY
ZHENKUN LI AND YI LIANG
Abstract.
In a recent paper, the first author and his collaborator developed a method tocompute an upper bound of the dimension of instanton Floer homology via Heegaard diagrams of3-manifolds. In this paper, for a knot inside S , we further introduce an algorithm that computesan upper bound of the dimension of instanton knot homology from knot diagrams. We test thealgorithm with all knots up to 7 crossings as well as a more complicated knot 10 . In the secondhalf of the paper, we show that if the instanton knot Floer homology of a knot has a specifiedform, then the knot must be an instanton L-space knot. Introduction
Knot theory is a central topic in low dimensional topology. In 1990, Floer introduced a knotinvariant called the instanton knot homology in [Flo90]. It has become a powerful tool in the studyof knot theory. For example, the instanton knot homology detects the genus and fibredness ofa knot, recovers the Alexander polynomial, and plays an important role in the establishment ofthe milestone result that Khovanov homology detects the unknot. See Kronheimer and Mrowka[KM10b, KM10a, KM11b, KM11a].The instanton knot homology of a knot K Ă S is a finite-dimensional complex vector spaceassociated to K . It is constructed by studying the solutions of sets of partial differential equationsover a closed oriented 3-manifold Y and the infinite cylinder R ˆ Y , where Y is obtained from theknot complement by attaching some standard piece of 3-manifold that also has a toroidal boundary.This nature of instanton knot homology makes it very difficult to compute. The breakthroughin computation was made by Kronheimer and Mrowka. Combining their work that the Eulercharacteristic of instanton knot homology recovers the Alexander polynomial of the knot, and thatthere exists a spectral sequence whose E page is the Khovanov homology and whose E page is theinstanton knot homology. Thus one obtains a lower bound of the dimension of the instanton knothomology via the Alexander polynomial and an upper bound via the Khovanov homology. Whenthese two bounds coincide, for example, for all alternating knots, the computation is done.Later, several groups of people studied the representation varieties of some special families ofknots to write down an explicit set of generators of the chain complex of the instanton knot homologyand thus obtained some better upper bounds than the one coming from Khovanov homology. See[HHK14, DS19, LZ20].Heegaard diagram is an effective combinatorial way to describe 3-manifolds and knots. In fact, anyknot can be described and is determined by its Heegaard diagrams. Since instanton knot homologyserves as a knot invariant, it is a priori determined by the Heegaard diagram of the knot. However,its construction through partial differential equations makes its relation to Heegaard diagramsvery implicit. To study this relation, recently, the first author and his collaborator established thefollowing result in [LY20]. a r X i v : . [ m a t h . G T ] F e b ZHENKUN LI AND YI LIANG
Theorem 1.1.
Suppose K Ă S is a knot, and p Σ , α, β q is a Heegaard diagram of K . Let H be ahandle body with B H “ Σ and γ Ă H is a (disconnected) oriented simple closed curve on H so thatthe following is true.(1) We have that γ has p g p Σ q ` q components.(2) We have that Σ z γ consists of two components of equal Euler characteristics.(3) We have that all β -curves are components of γ .Then we have the following inequality. dim C KHI p K q ď dim C p H, γ q . Here,
KHI is the instanton knot homology of K , and SHI is the sutured instanton Floer homologyof the balanced sutured manifold p H, γ q . Sutured instanton Floer homology associates a finite-dimensional complex vector space to everybalanced sutured manifold. It was introduced by Kronheimer and Mrowka in [KM10b]. Thecomputation of
SHI p M, γ q for a general balanced sutured manifold p M, γ q is also difficult, due tothe same reason as KHI . However, in the special case when M “ H is a handle body, the firstauthor and his collaborator developed an algorithm to compute an upper bound of SHI p H, γ q in[GL19]. So, equipped with Theorem 1.1, one can obtain an upper bound on the dimension of theinstanton knot homology of a knot from any Heegaard diagram of that knot. Following this idea, Liand Ye were able to compute a conjecturally sharp upper bound for all p , q -knots in [LY20].In this paper, we utilize Theorem 1.1 further and obtain the following. Theorem 1.2.
Suppose K Ă S is a knot. Let D be any knot diagram of K . Then there is analgorithm to compute an upper bound of the dimension of KHI p S , K q out of D .Remark . The project in the current paper was launched right after the completion of [LY20],where the first author and his collaborator first introduced Theorem 1.1. Later, Theorem 1.1 wasfurther used to prove the main result of [BLY20]. By [BLY20, Theorem 1.1], one knows that thedimension of
KHI p S , K q is bounded by the number of generators of { CF K p S , K q , where { CF K is the chain complex of the hat version of knot Floer homology introduced by Ozsv´ath and Szab´o[OS04]. Since the number of generators of { CF K p S , K q can also be computed directly from anygiven diagram of the knot K , [BLY20, Theorem 1.1] gives rise to an algorithm that is differentfrom the one in Theorem 1.2. Though we haven’t compared the effectiveness between these twoalgorithms.We further testify the algorithm in Theroem 1.2 with knots of crossing numbers at most 7. Wefound that all the bounds from the algorithm are sharp. We also test a more complicated knot 10 .See Table 1 for more details.It is worth mentioning that all knots of crossing number at most 7 are alternating. So thedimensions of KHI for them have been known due to the work of Kronheimer and Mrowka [KM10a,KM11a] as discussed above. However, the upper bounds coming from Khovanov homology dependson the establishment of a spectral sequence that relates instanton knot homology with Khovanonvhomology. So Theorem 1.2 provides an alternative proof that is independent of Kronheimer andMrowka’s spectral sequence.Instanton L-space knots are those knots inside S that admits a Dehn surgery whose instantonFloer homology has minimal dimension. As mentioned above, the computation of instanton Floerhomology is a big open problem, so in general, it is hard to identify instanton L-space knots. In thispaper, we give a sufficient condition in terms of the instanton knot homology. OME COMPUTATIONS ON INSTANTON KNOT HOMOLOGY 3
Theorem 1.4.
Suppose K Ă S is a knot of genus g . Suppose further that (1.1) KHI p K, i q – " C , | i |“ g, g ´ , , Otherwise Then K admits an instanton L-space surgery. Combined with the results from [LPCS20, BS20], we conclude the following.
Corollary 1.5.
Suppose K Ă S is a knot of genus g whose instanton knot homology is describedas in (1.1), then one and exactly one of the following two statements is true.(1). S g ´ p K q is an instanton L-space, and for any rational number r “ pq with q ě , we have (1.2) dim C I p S r p K qq “ " p if r ě g ´ p g ´ q ¨ q ´ p otherwise (2). S ´ g p K q is an instanton L-space, and for any rational number r “ pq with q ě , we have (1.3) dim C I p S r p K qq “ " ´ p if r ď ´ g p g ´ q ¨ q ` p otherwise Acknowledgement
The authors would like to thank the PCP program for making this collabo-ration possible. 2.
Preliminaries
Definition 2.1 ([Juh06, KM10b]) . A balanced sutured manifold p M, γ q consists of a compactoriented 3-manifold M with non-empty boundary together with a closed 1-submanifold γ on B M .Let A p γ q “ r´ , s ˆ γ be an annular neighborhood of γ Ă B M and let R p γ q “ B M z int p A p γ qq . Theysatisfy the following properties.(1) Neither M nor R p γ q has a closed component.(2) If B A p γ q “ ´B R p γ q is oriented in the same way as γ , then we require this orientationof B R p γ q induces one on R p γ q . The induced orientation on R p γ q is called the canonicalorientation .(3) Let R ` p γ q be the part of R p γ q so that the canonical orientation coincides with the inducedorientation on B M , and let R ´ p γ q “ R p γ qz R ` p γ q . We require that χ p R ` p γ qq “ χ p R ´ p γ qq .If γ is clear in the contents, we simply write R ˘ “ R ˘ p γ q , respectively. Theorem 2.2 (Kronheimer and Mrowka [KM10b]) . For any balanced sutured manifold p M, γ q , wecan associate a finite-dimensional complex vector space, which we denote by SHI p M, γ q , to p M, γ q .It serves as a topological invariant of the pair p M, γ q . Definition 2.3 (Kronheimer and Mrowka [KM10b]) . For knot K Ă S , define its instanton knothomology , which we denote by KHI p K q , to be KHI p K q “ SHI p S p K q , γ µ q , where γ µ consists of two meridians of K .For a connected closed oriented 3-manifold Y , define its framed instanton Floer homology ,which is denoted by I p Y q , to be I p Y q “ SHI p Y p q , δ q , where Y p q “ Y z D is obtained from Y by removing a 3-ball and δ Ă B Y p q is a connected simpleclosed curve. ZHENKUN LI AND YI LIANG
Definition 2.4.
A knot K Ă S is called an instanton L-space knot if there exists a non-zerointeger n so that dim C I p S n p K qq “ | n | . Here S n p K q is the three manifold obtained from S by an n surgery along the knot K .In [KM10b, KM10a], Kronheimer and Mrowka studied many basic properties of KHI , which wesummarize as in the following two theorems.
Theorem 2.5.
Suppose K Ă S is a knot of genus g , then the following is true.(1) There is a Z -grading on KHI p K q , which is called the Alexander grading : KHI p K q “ à i P Z KHI p K, i q . (2) For any i P Z with | i |ą g , we have KHI p K, i q “ .(3) We have KHI p K, g q ‰ .(4) For any i P Z , we have KHI p K, i q –
KHI p k, ´ i q .(5) The knot K is fibred if and only if KHI p K, g q – C . Theorem 2.6 (Kronheimer and Mrowka [KM10a]) . Suppose K Ă S is a knot. Let ∆ K p t q “ ÿ i P Z a i t i be its Alexander polynomial. Then we know that dim C KHI p K q ě ÿ i P Z | a i | . Here | ¨ | means the abstract value.
In [Li19a, Li19b, LY20], the first author and his collaborators studied different sutures on theknot complements. Suppose K Ă S is a knot. Let γ p p,q q be the suture on B S p K q consisting of twosimple closed curves of slope q { p on B S p K q . We have the following. Theorem 2.7.
Suppose K Ă S is a knot of genus g . For any pair of co-prime integers p p, q q P Z , SHI p S p K q , γ p p,q q q admits a grading that sits in either Z or Z ` :If q is odd, then SHI p S p K q , γ p p,q q q “ à i P Z SHI p S p K q , γ p p,q q , i q . If q is even, then
SHI p S p K q , γ p p,q q q “ à i P Z ` SHI p S p K q , γ p p,q q , i q . Furthermore, the following is true.(1) For any i with | i |ą g ` q ´ , we have SHI p S p K q , γ p p,q q , i q “ (2) We have SHI p S p K q , γ p p,q q , g ` q ´ q “ . (3) For any i , we have SHI p S p K q , γ p p,q q , i q – SHI p S p K q , γ p p,q q , ´ i q . OME COMPUTATIONS ON INSTANTON KNOT HOMOLOGY 5 (4) We have (2.1)
SHI p S p K q , γ p , ´ g ´ q , q – C . (5) We have (2.2) I p S ´ g ´ p K qq – g à i “´ g SHI p S p K q , γ p , ´ g ´ q , i q . Bypass triangles were introduced in the instanton theory by Baldwin and Sivek [BS18] to relatedifferent sutures on the knot complements. The first author further studied a graded version ofbypass exact triangle in [Li19a].
Theorem 2.8.
Suppose K Ă S is a knot. For any i P Z , there are three exact triangles (2.3) SHI p S p K q , γ p , ´ g q , i ` q ψ ` ,i (cid:47) (cid:47) SHI p S p K q , γ p , ´ g ´ q , i q (cid:115) (cid:115) KHI p K, i ´ g q (cid:79) (cid:79) (2.4) SHI p S p K q , γ p , ´ g q , i ´ q ψ ´ ,i (cid:47) (cid:47) SHI p S p K q , γ p , ´ g ´ q , i q (cid:115) (cid:115) KHI p K, i ` g q (cid:79) (cid:79) (2.5) SHI p S p K q , γ p , ´ g q , g ` i ` q (cid:47) (cid:47) SHI p S p K q , γ p , ´ g ´ q , i q (cid:115) (cid:115) SHI p S p K q , γ p , ´ g ´ q , ´ g ` i q (cid:79) (cid:79) One of the main result of [BS18] can be re-stated as follows.
Theorem 2.9.
Suppose K is a fibred knot and is not right veering. Then the map ψ ` , g ´ : SHI p S p K q , γ p , ´ g q , g ´ q Ñ SHI p S p K q , γ p , ´ g ´ q , g ´ q in (2.3) is zero. Next, we introduce the Heegaard diagrams of 3-manifolds and knots.
Definition 2.10. A (genus g ) diagram is a triple p Σ , α, β q so that the followings hold.(1) We have Σ being a connected closed surface of genus g .(2) We have α “ t α , . . . , α m u and β “ p β , . . . , β n q being two sets of pair-wise disjoint simpleclosed curves on Σ. We do not distinguish the set and the union of curves.A (genus g ) Heegaard diagram is a (genus g ) diagram p Σ , α, β q satisfying the followingconditions.(1) We have | α | “ | β | “ g , i.e. , there are g many curves in either tuple.(2) The complements Σ z α and Σ z β are connected. ZHENKUN LI AND YI LIANG
It is a basic fact in low dimensional topology.
Theorem 2.11.
Any knot K Ă S admits a Heegaard diagram. Knots diagrams and instanton knot homology
In this section, we prove Theorem 1.2.
Proof of theorem 1.2.
Suppose K Ă S is a knot and D is a knot diagram of the knot. Step 1 . We construct a Heegaard diagram from a knot diagram. To do this, we first form thesingular knot K s by replacing every crossing of D with two arcs that intersect at one point. We canthink of K s as embedded in S . Then let H “ S z N p K s q . It is straightforward to check that H is ahandle body. Let Σ “ B H and the α -curves consists of g “ c p D q ` H , where g is thegenus of H and c p D q denotes the number of crossings in D . Next, we need to find the β -curves. Weneed g many of them. We draw one β -curve around each crossing of K according to the principleshown in Figure 1 (note there are c p K q “ g ´ K to be the last β -curve. As in [Hom20], this gives us a Heegaard diagram of K . Figure 1.
The red curves on the right are the β -curves. Step 2 . We construct a sutured handle body from p Σ , α, β q . We simply pick H “ S z N p K s q to be the handle body, and all β -curves are the components of γ . Also, γ has one last componentobtained by p g ´ q band sums on β -curves that make all g many β -curves into one connected simpleclosed curve that can be isotoped to be disjoint from all of the original β -curves. Theorem 1.1 thenapplies and we have dim C KHI p K q ď dim C SHI p H, γ q . OME COMPUTATIONS ON INSTANTON KNOT HOMOLOGY 7
Step 3 . We compute an upper bound of dim C SHI p H, γ q . This is done by induction based onthe following two lemmas.To present the first lemma, recall we have g many α -curves. Call them α ,..., α g . Lemma 3.1 (Kronheimer and Mrowka [KM10b]) . If for every index i P t , ..., g u , we have | α i X γ | ď , where | ¨ | denotes the number of intersection points, then dim C SHI p H, γ q ď . Lemma 3.2 (Baldwin and Sivek [BS18]) . Suppose i P t , ..., g u and β Ă B D i is part of B D i so that B β Ă γ and | β X γ | “ . See Figure 2 for an example. Within a neighborhood of β , we can alter the suture γ as shown inFigure 2, and get two new sutures γ and γ . Then we have dim C SHI p H, γ q ď dim C SHI p H, γ q ` dim C SHI p H, γ q . It is clear that | γ X D i | ď | γ X D i | ´ | γ X D i | ď | γ X D i | ´ . So using Lemma 3.1 we can reduce the number of intersections of γ with arbitrary meridian disk of H . When γ intersects all meridian disks at most two times, Lemma 3.2 applies and we can obtain abound on dim C SHI p H, γ q for any suture γ . a b cdef γ a b cdef γ a b cdef γ β Figure 2.
A by-pass move obtaining γ and γ from γ . The red curves are thesutures. The dotted circle bounds the disk E Ă Σ n . (cid:3) ZHENKUN LI AND YI LIANG
Knots Upper bound for dim C KHI
Alexander polynomial3 t ´ ` t ´ ´ t ` ´ t ´ t ´ t ` ´ t ´ ` t ´ t ´ ` t ´ ´ t ` ´ t ´ ´ t ` t ´ ` t ´ ´ t ´ t ´ t ` ´ t ´ ` t ´ t ´ t ` t ´ ` t ´ ´ t ´ ` t ´
11 3 t ` ` t ´
13 2 t ´ t ` ´ t ´ ` t ´
15 4 t ´ ` t ´
17 2 t ´ t ` ´ t ´ ` t ´ ´ t ` t ´ ` t ´ ´ t ´ t ´ t ` ´ t ´ ` t ´ t ´ t ´ t ` ´ t ´ ´ t ´ ` t ´ Table 1.
Knots with small crossings. We use Rolfsen’s knot table in [Rol90] toname all these knots. To obtain a lower bound (which coincide with the upperbound), one can sum up the abstract value of all coefficients of the Alexanderpolynomial.We have performed computations for all knots with crossing number at most 7, as well as a morecomplicated knot 10 . The results are summarized in Table 1.
Remark . Besides knots with small crossings, we also tried another knot 10 . The reason whywe work on this particular knot is the following. As explained in the introduction, for alternatingknots, the upper bounds of the dimension of
KHI coincide with the lower bound coming from theAlexander polynomial. In [LY20], the first author and his collaborator computed upper bounds forall p , q -knots, and for many families of p , q -knots, the upper bounds obtained in [LY20] are betterthan those from Khovanov homology. Hence we are interested in finding more examples outside therange of alternating knots and p , q -knots. We didn’t find a complete list for all p , q -knots, so weturn to search in the knots with tunnel number at least 2, since all p , q -knots are known to havetunnel number 1. So 10 is the first knot K came into our sight that satisfies the following threeconditions:(1) The knot has tunnel number at least 2 and is not alternating.(2) The upper bound from Khovanov homology is strictly larger than the lower bound from theAlexander polynomial.(3) The Alexander polynomial of the knot is not too complicated.Unfortunately, the upper bound we obtained for 10 , which is 17, coincides with the upperbound from Khovanov homology. Note this upper bound is strictly greater than the lower boundfrom Alexander polynomial, so the precise dimension of KHI for 10 is still open.4.
Dehn surgeries on knots
In this section, we prove Theorem 1.4.
OME COMPUTATIONS ON INSTANTON KNOT HOMOLOGY 9
Proof of Theorem 1.4.
Suppose K Ă S is a knot of genus g and its instanton knot homology satisfiesthe assumption in the hypothesis of the theorem. Note from the assumption that KHI p K, g q – C and Theorem 2.5, we know that K is fibred and g ě
2. Then either K or the mirror of K is notright veering. By passing to its mirror if necessary, we can assume that K itself is not right veering.We begin with a few lemmas. Lemma 4.1.
We have (4.1)
SHI p S p K q , γ p , ´ g ´ q , g q – C and (4.2) SHI p S p K q , γ p , ´ g q , g ´ q – C . Proof.
Applying term 2 of Theorem 2.5, Formula (2.3) from Theorem 2.8, and the assumptionthat
KHI p K, g q – C in the hypothesis, we conclude (4.1). Similarly, (4.2) follows from term 2 ofTheorem 2.5, Formula (4.1), and Formula (2.3) from Theorem 2.8. (cid:3) Lemma 4.2.
SHI p S p K q , γ p , ´ g ´ q , g ´ q “ .Proof. Note we have argued that K is fibred and also assumed that it is not right veering. HenceTheorem 2.9 applies. Then the lemma follows from Theorem 2.9, Formula (4.2), the assumptionthat KHI p K, g ´ q – C , and Theorem 2.8. (cid:3) Lemma 4.3.
We have
SHI p S p K q , γ p , ´ g ´ q , i q – SHI p S p K q , γ p , ´ g q , i ´ q for i ą .Proof. Since ´ i ´ g ă ´ g , we have KHI p K, ´ i ´ g q “ SHI p S p K q , γ p , ´ g ´ q , ´ i q – SHI p S p K q , γ p , ´ g q , ´ i ` q . By term 3 of Theorem 2.7, we have an isomorphism(4.4)
SHI p S p K q , γ p , ´ g ´ q , ´ i q – SHI p S p K q , γ p , ´ g ´ q , i q and(4.5) SHI p S p K q , γ p , ´ g q , ´ i ` q – SHI p S p K q , γ p , ´ g q , i ´ q The lemma then follows after substituting in Equation (4.3). (cid:3)
Lemma 4.4.
We have
SHI p S p K q , γ p , ´ g ´ q , i q “ " g ` ď i ď g ´ C ď i ď g Proof.
To start, in Lemma 4.2, we have proved that
SHI p S p K q , γ p , ´ g ´ q , g ´ q – . By Lemma 4.3, we know that
SHI p S p K q , γ p , ´ g q , g ´ ´ q – SHI p S p K q , γ p , ´ g ´ q , g ´ q “ . Since
KHI p K, g ´ q “ i “ g ´ SHI p S p K q , γ p , ´ g ´ q , g ´ q – SHI p S p K q , γ , ´ g , g ´ ´ q “ . Repeating the above argument once more for i “ g ´
3, we conclude that
SHI p S p K q , γ p , ´ g ´ q , g ´ q – . We can keep running this argument until we finish the case i “ g `
1, where we have(4.6)
SHI p S p K q , γ p , ´ g ´ q , g ` q “ . By far we have proved the vanishing part of the lemma. For the other half of the lemma, we usea similar argument. Note we have
KHI p K, q – C by the hypothesis of the theorem, and SHI p S p K q , γ p , ´ g q , g ` q – i “ g in Formula (2.3), we conclude SHI p S p K q , γ p , ´ g ´ q , g q – C . Then using a similar repetitive argument as above, we conclude SHI p S p K q , γ p , ´ g ´ q , i q – C for 2 ď i ď g . (cid:3) Lemma 4.5.
For any i P Z such that ´ g ď i ď g , we have SHI p S p K q , γ p , ´ g ´ q , i q – C Proof.
First, taking i “ g , by term 1 of Theorem 2.7, we know that SHI p S p K q , γ p , ´ g q , g ` i ` q “ SHI p S p K q , γ p , ´ g q , g ` q “ . Also, by term 4 of Theorem 2.7, we have
SHI p S p K q , γ p , ´ g ´ q , ´ g ` i q “ SHI p S p K q , γ p , ´ g ´ q , q – C . Then by Formula (2.5) in Theorem 2.8, we have
SHI p S p K q , γ p , ´ g ´ q , g q – SHI p S p K q , γ p , ´ g ´ q , ´ g q – C . Next, we skip the case i “ g ´ i so that g ´ ě i ě ´ g ` . The case i “ g ´ i P r , g ´ s , we have SHI p S p K q , γ p , ´ g ´ q , g ` i ` q “ . OME COMPUTATIONS ON INSTANTON KNOT HOMOLOGY 11
Also, by Lemma 4.3, we have
SHI p S p K q , γ p , ´ g q , g ` i ` q – SHI p S p K q , γ p , ´ g ´ q , g ` i ` q . Thus, we derive
SHI p S p K q , γ p , ´ g q , g ` i ` q “ i P r , g ´ s .On the other hand, by term 3 of Theorem 2.7, we have an isomorphism SHI p S p K q , γ p , ´ g ´ q , i ´ g q – SHI p S p K q , γ p , ´ g ´ q , g ´ i q , and by Lemma 4.4, for any i P r , g ´ s , we have SHI p S p K q , γ p , ´ g ´ q , g ´ i q – C . Thus, by Formula (2.5), we have
SHI p S p K q , γ p , ´ g ´ q , i q – C. Similarly, for ´ g ` ď i ď ´
1, by Lemma 4.4, we have
SHI p S p K q , γ p , ´ g ´ q , g ´ i q – , and SHI p S p K q , γ p , ´ g q , g ` i ` q – SHI p S p K q , γ p , ´ g ´ q , g ` i ` q – C . Then, by Formula 2.5, we conclude
SHI p S p K q , γ p , ´ g ´ q , i q – C for g ´ ě i ě ´ g ` ´ g ď i ď g except i “ g ´
1, which could be reached by term 3in Theorem 2.7 and the conclusion above since
SHI p S p K q , γ p , ´ g ´ q , g ´ q – SHI p S p K q , γ p , ´ g ´ q , ´ g q – C The lemma then follows. (cid:3)
Theorem 1.4 then follows directly from Lemma 4.5 and term 5 of Theorem 2.7. (cid:3)
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Department of Mathematics, Stanford University
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