A Combinatorial Description of the Knot Concordance Invariant Epsilon
AA COMBINATORIAL DESCRIPTION OF THE KNOTCONCORDANCE INVARIANT EPSILON
SUBHANKAR DEY AND HAKAN DOĞA
Abstract.
In this paper, we give a combinatorial description of the concordance in-variant ε defined by Hom in [5], prove some properties of this invariant using grid ho-mology techniques. We also compute ε of ( p, q ) torus knots and prove that ε ( G + ) = 1 if G + is a grid diagram for a positive braid. Furthermore, we show how ε behaves under ( p, q ) -cabling of negative torus knots. Introduction
While a significant progress has been made in recent years, it still remains largelyopen in low-dimensional topology and knot theory to understand the structure of theknot concordance group. Concordance invariants obtained from various versions of knotFloer homology have proved to be very efficient in further analyzing and understandingthe structure of the concordance group. Ozsváth and Szabó defined the concordanceinvariant τ in [10] and prove that τ gives a lower bound for the 4-ball genus of a knot K . In the same paper, they prove several useful properties of τ and compute τ forcertain family of knots. Another important concordance invariant Υ was defined in [14]by Stipsicz, Ozsváth and Szabó. Using Υ , they prove that C T S , which is the subgroupof the smooth concordance group generated by topologically slice knots, contains a Z ∞ summand. Finally, we turn our focus to another concordance invariant ε that is definedusing CF K ∞ ( K ) by Hom in [5]. In [6], Hom uses ε to calculate the τ invariant of ( p, q ) torus knots and ( p, q ) cables of a knot K .Our main goal in this paper is to provide a combinatorial description for ε usinggrid homology, compute its value for ( p, q ) torus knots and positive braids, and finallyexamine its behavior under ( p, q ) cabling on torus knots. In an upcoming work, we generalize ε for knots in lens spaces using the grid descriptiondefined by Baker-Grigsby-Hedden in [1]. We will also examine the effect of ( p, q ) -cablingon ε for any non-trivial knot K ⊂ S and prove similar results for K ⊂ L ( p, q ) .The combinatorial treatment of knot Floer homology, namely grid homology, proved tobe a valuable tool, especially for computational purposes. Initially defined by Manolescu-Ozsváth-Sarkar in [7], it was used to give a simpler proof of the Milnor conjecture [13],to study transverse and Legendrian knots in S ([12], [8]), as well as to compute theconcordance invariant τ for several knots and knot families [2]. Following this idea, wewill use grid diagrams to give a description for the concordance invariant ε . Let G denotea grid diagram for a knot K , then the first main result can be stated as follows; This work is supported in part by a Simons Foundation grant No. 519352. It is important to note here that these results, except for the positive braids, were proved by Homin [6]. We reprove these results only using grid homology. a r X i v : . [ m a t h . G T ] O c t DEY AND DOĞA
Theorem 1.1. ε ( G ) defined via grid homology is a concordance invariant, in the sensethat it satisfies the following properties;a) If K is slice, then ε ( G ) = 0 ,b) ε ( (cid:57) G ) = (cid:57) ε ( G ) where (cid:57) G denotes the diagram for the mirror reverse of K ,c) If ε ( G ) = ε ( G ) for some knots K and K , then ε ( G G ) = ε ( G ) .d) If ε ( G ) = 0 , then ε ( G G ) = ε ( G ) . After establishing the above properties which immediately imply that this is a con-cordance invariant, we return to some computations. We have the following result fornegative torus knots in S , Theorem 1.2.
Let G − p,q denote the standard grid diagram for the negative ( − p, q ) -torusknot with grid index equal to | p | + | q | . Then; ε ( G − p,q ) = − q > | q | = 11 q < − For the theorem above, we will use the standard grid diagram for negative torus knotsused in [13] in the proof of Milnor conjecture. In [6], bordered Heegaard Floer homologyis used to prove the behaviour of ε under ( p, q ) -cabling. We will use grid homology toprove a similar result, yet for only ( p, q ) -cables of negative torus knots. Theorem 1.3.
Let G denote the grid diagram for ( − p (cid:48) , q (cid:48) ) torus knot and let K p,q be agrid diagram for a ( p, q ) -cable of G . Then; • If ε ( G ) (cid:54) = 0 , then ε ( K p,q ) = ε ( G ) . • If ε ( G ) = 0 , then ε ( K p,q ) = ε ( G p,q ) . Furthermore, we extend our results to positive braids and show that any positive braidhas epsilon equal to 1. This result relies on positive braids being fibered which is provedby Stallings in more generality in [16], for all homogeneous braids, along with Hedden’sresult in [4] where he proves τ ( K ) = g for fibered, strongly quasipositive knots in S . Wespecifically appeal to Proposition 2.1 in [4] and equivalent statements provided there.As a result, we have the following theorem, Theorem 1.4.
Let G + denote the grid diagram for a positive braid B , then ε ( G + ) = 1 . This paper is structured in the following way; in Section 2, we revisit some definitionsfrom grid homology and describe the method that allows us to compute ε using gridhomology. Section 3 contains the proof of Theorem 1.1 as well as the method to constructthe connected sum of knots using grid diagrams. In Section 4, we compute the value of ε for negative torus knots and show how ε behaves under ( p, q ) -cabling. In Section 5,we describe how we can view grid diagrams to represent braids and show that ε = 1 forpositive braids. Acknowledgements.
The authors would like to thank Çağatay Kutluhan, András Stip-sicz, John Etnyre and Lenhard Ng for their comments and suggestions on the first draftof the paper. We are especially thankful to Jen Hom for suggesting to look into positivebraids and for her helpful comments. The first author would also like to acknowledge
COMBINATORIAL DESCRIPTION OF THE KNOT CONCORDANCE INVARIANT EPSILON 3 the support of the Dissertation Fellowship awarded by the Department of Mathematicsat University at Buffalo during the project.2.
Preliminaries on ε ( K ) and Grid Homology: Grid homology was defined by Manolescu-Ozsváth-Sarkar in [7] for knots in S , usingthe nice diagrams introduced by Sarkar-Wang in [15]. As we will discuss further, themain advantage of grid homology is the simplification of the domains counted by thedifferential and the combinatorial nature of its setup, despite increasing the complexityof the knot Floer complex. Many different properties of the Grid Homology settingare discussed and extended in [13] and we will be following the notation conventionsdescribed there. Before we define the combinatorial recipe for ε ( K ) , we will briefly recallthe grid homology setup. Definition 2.1.
A grid diagram G for a knot K in S is given by the following data; • n × n grid decorated with a set of n X-markings and a set of n O-markings thatwe denote by X and O , respectively. • Each row and column contains exactly one O and one X marking. • No double markings inside the small squares.To specify K , we draw vertical strands between the markings and orient them fromX to O, and then draw the horizontal ones orienting the strands from O to X. We alsofollow the convention that vertical strands go over horizontal ones. This gives us thegrid diagram G of K with a specified orientation. We call the grid size n the grid index.Furthermore, we identify the top and bottom edges, right-most and left-most edgesof the grid diagram, and consider the toroidal grid diagram along with the standardcounterclockwise orientation on the torus.A grid state x is a one-to-one correspondence between horizontal and vertical circleson the grid torus, equivalently an n-tuple x = { x , x , . . . , x n } where each x i comes fromthe intersection points of horizontal and vertical circles. The set of grid states is denotedby S ( G ) . Now given x , y ∈ S ( G ) such that they differ at exactly n − coordinates, wesay r is an empty rectangle from x to y and denote the set of empty rectangles by r ∈ Rect ( x , y ) if; • Int ( r ) ∩ x = Int ( r ) ∩ y = ∅• ∂ ( ∂ h ) = y − x and ∂ ( ∂ v ) = x − y where ∂ h and ∂ v denote horizontal and verticalboundaries respectively.For our purposes, we will focus on the versions of the grid homology that are referredas filtered grid complex and collapsed grid homology for links in [7], denoted by GC − ( G ) and cGH − ( G ) respectively. We slightly modify the boundary map of GC − ( G ) and reduceit to a single U -action as follows; ∂ − ( x ) = (cid:88) y ∈ S ( G ) (cid:88) { r ∈ Rect ( x , y ) } U | r ∩ O | y DEY AND DOĞA
This complex is also equipped with two gradings, Maslov and Alexander grading.Both of these gradings can be calculated by a function on the planar realization of thegrid diagram. The candidate function is constructed in [13] and calculates the Maslovgrading as follows for x ∈ S ( G ) ; M O ( x ) = J ( x , x ) − J ( x , O ) + J ( O , O ) + 1 where J ( A, B ) is a symmetrization of the function I ( A, B ) which counts the numberof elements of B that lie in the North-East of the elements of A . Using this, we can alsodefine the following Alexander grading function; A ( x ) = 12 ( M O ( x ) − M X ( x )) − ( n −
12 ) where n is the grid index. O O O O OX X X X X
Figure 1.
Grid diagram for the left-handed trefoil (LHT), two grid states x and y are given by empty and full circles respectively. Darker shadedrectangle and lighter shaded rectangle are two rectangles from x to y .Notice that darker shaded rectangle is counted by the differential whereaslighter shaded one is not since it is not an empty rectangleNow we can also define the concordance invariant τ which will be helpful for our dis-cussions about ε . Since our aim is to provide a combinatorial description, we follow thedefinition from grid homology which is equivalent to the definition in the holomorphictheory. For a given knot K , let G be a grid diagram of it. We have the following defini-tion; τ ( K ) = - max{ n | there exists a homogeneous, non-torsion element in GH − ( G ) withAlexander grading equal to n }We briefly recall the definition of ε as in [5]. We start by considering ( CF K ∞ ( K ) , ∂ ∞ ) and then restrict ourselves to "vertical" and "horizontal" homologies. The invariant ε is defined through the relation between these two homologies. More precisely, let [ x ] be COMBINATORIAL DESCRIPTION OF THE KNOT CONCORDANCE INVARIANT EPSILON 5 the generator of H ∗ ( C { i = 0 } ) . We can view [ x ] as a class in the horizontal complex H ∗ ( C { j = τ } ) which yields the following definition of ε ; • ε ( K ) = 0 if and only if [ x ] is in the kernel, but not in the image of the horizontaldifferential. • ε ( K ) = 1 if and only if [ x ] is in the image of the horizontal differential. • ε ( K ) = − if and only if [ x ] is not in the kernel of the horizontal differential.Now, we can give the combinatorial description of ε using grid diagrams. Let G bea grid diagram of the knot K . Recall that GH − ( K ) /T ors (cid:39) F [ U ] . Let [ x τ ] be thegenerator of the free U -module part of GH − ( K ) . Notice that this implies − A ( x τ ) = τ ( K ) . To define ε , we have to observe how this homology class interacts with the horizontal complex in our context. We observe that this horizontal complex can beobtained by reversing the roles of i and j , filtration with respect to U powers andAlexander filtration respectively, and then also reversing all the arrows in the CF K ∞ ( K ) complex. Using the symmetry properties of the grid homology (Propositions 7.1.1, 7.1.2and 7.4.3 in [13]), this can be achieved by reflecting the grid diagram G with respect toa horizontal axis and then reversing all X and O markings. This will yield a diagram for − K , that is the reverse of the mirror of K . Let (cid:57) G be the grid diagram obtained via this.Now, the horizontal complex is equipped with the horizontal differential that preservesthe Alexander filtration and also counts the rectangles that have non-zero U -multiplicity.We denote this differential by ∂ −− G and define it as follows; ∂ −− G ( x ) = (cid:88) y ∈ S ( G ) (cid:88) { r ∈ Rect ( x , y , ) , r ∩ X = ∅ , | r ∩ O |(cid:54) =0 } U | r ∩ O | y The above differential is the combinatorial version of the horizontal differential de-scribed in [5]. Hence, we have the following definition of ε that is computed using griddiagrams. Definition 2.2.
Let G and (cid:57) G be grid diagrams for K and (cid:57) K , and [ x τ ] be the generatorof the free part of GH − ( G ) , then we have; • ε ( G ) = 0 if and only if [ x τ ] ∈ Ker( ∂ −− G ), but [ x τ ] (cid:54)∈ Im( ∂ −− G ). • ε ( G ) = 1 if and only if [ x τ ] (cid:54)∈ Ker( ∂ −− G ). • ε ( G ) = − if and only if [ x τ ] ∈ Im( ∂ −− G ).Note that to be consistent with the definition of ε invariant in knot Floer homology,the combinatorial definition we outline here multiples the invariant with -1 . This isdone since our description uses grid diagrams and the combinatorial definition of theunblocked grid homology. A similar strategy is used in [13, Remark 6.1.6] to define τ inthe grid setting. Furthermore, we use ε ( G ) to distinguish from ε ( K ) and indicate thatthe computations are made using the grid techniques we describe in this paper. DEY AND DOĞA
Figure 2.
From left to right, we start with the standard grid diagramfor the unknot G O , reflect it with respect to a horizontal axis and thenchange X and O markings to obtain (cid:57) G O In Figure [2], we see the example of the unknot G O . Notice that the grid state givenby full circles is a cycle since both rectangles leaving this grid state contain X markings,and this cycle generates the vertical homology. After applying the above definition tocompute ε , we can immediately see that in (cid:57) G O , the same element is non-trivial. Hence,we can conclude that ε ( G O ) = 0 .Our goal is to prove that this combinatorial method described above provides a pow-erful and efficient way to compute ε . To prove the Theorems 1.2 and 1.3, we use thestandard grid diagram for ( − p, q ) -torus knots.To this end, we briefly recall the standard method of encoding any grid diagramusing permutations. In the planar realization of a grid diagram, we enumerate thecolumns starting from left and enumerate the rows starting from the bottom, then wecan represent the grid diagram using two permutations that are elements of S n if thegrid index is n . These permutations are denoted by σ X and σ O recording the positionsof X and O -markings. We follow the convention that if a marking is in the i th columnand j th row, then the permutation takes i to j , and for simplicity, we only write j in thepermutation.In the case of a ( − p, q ) -torus knot, we can form a grid diagram denoted by G − p,q with grid index n = | p | + | q | and with permutations σ O = (1 , , , . . . , p + q ) and σ X =( p + 1 , p + 2 , . . . , p + q, , , . . . , p ) . See Figure [3] for an example.3. Concordance Properties of ε In this section, we prove that ε is a knot concordance invariant. We make use of thecombinatorial theory and some results from there to prove the properties of ε listed inTheorem 1.1.Since we also need to consider connected sum of knots using grid diagrams, we definea method of constructing the grid diagram of connected sum of knots. This methodallows us to carry the information of the individual knots to the larger diagram of theconnected sum.Let G (cid:48) and G (cid:48) be two grid diagrams of the knots K and K with grid indices n and m respectively. By the definition of grid diagrams, each row and column contain an X -marking. In G (cid:48) , cyclically permute the columns such that the X -marking in the bottomrow is in the left corner, and in G (cid:48) , cyclically permute the columns as well such that the COMBINATORIAL DESCRIPTION OF THE KNOT CONCORDANCE INVARIANT EPSILON 7
O O O O OX X X X X
Figure 3.
Grid diagram G − , for the left-handed trefoil, grid index is 5and σ O = (1 , , , , and σ X = (3 , , , , XX X XX X XX X X X XX
OO OO OOOO O O OOO
XX X XX X XX X X X XX
OO OO OOOO O O OOO
Figure 4.
Two schematic diagrams showing how to form the connectedsum from the disjoint union, on the left a grid diagram for the disjointunion of a link and the unknot, on the right a grid diagram for theirconnected sum X -marking in the top row moves to the right corner. Call the resulting diagrams G and G . After this step, place G to upper-right and G in the lower-left corner diagonally,and form a new grid diagram with grid index equal to n + m . Notice that this gives agrid diagram for K (cid:116) K . To produce a grid diagram for K K , we interchange the O -markings in the first column of G and the last column of G in the new diagram,cf. Figure [4], while keeping the X -markings fixed. As discussed in [13], this is a gridrealization of a saddle move which performs the connected sum operation. We denote theresulting diagram G G . Furthermore, a grid state in G G has n + m intersectionpoints. Hence, if x ∈ S ( G ) and x ∈ S ( G ) , we use x ∪ x to denote the union of theircoordinates on the resulting grid diagram. They will naturally produce a grid state in G G since top and right-most edges are omitted on the toroidal grid diagrams whenspecifying the grid states. DEY AND DOĞA
It is known in the literature that for any knot K in S , K (cid:57) K is slice and as aresult, concordance invariants such as ε vanish. Another advantage of the connectedsum construction using grid diagrams is that we can directly see that ε ( G (cid:57) G ) = 0 where G is a grid diagram of K . As shown schematically in Figure 5, we start bypositioning the grid diagrams of K and (cid:57) K . To compute ε , we follow our constructionand reflect the diagram with respect to a horizontal axis and change the markings. Weimmediately see that, we obtain a diagram for K in the upper-left and a diagram for (cid:57) K in the lower-right of the diagram. By cyclically permuting one of G or (cid:57) G , we observethat we go back to the original diagram. This means that the homology class [ x ] thatwe keep track of to compute ε is mapped to itself in the final diagram. As a result, it isstill the generator of the free part of GH − in the resulting diagram, which implies that ε ( G (cid:57) G ) = 0 . Figure 5.
Schematic picture for the connected sum, reflected and cycli-cally permutedNotice that this construction increases the complexity of the chain complex, that isif G and G has n ! and m ! generators respectively, the connected sum has ( n + m )! generators. However, we have the following lemma which tells us that we can transferthese grid states preserving the additivity of Alexander grading. Lemma 3.1.
Given two grid diagrams G and G for K and K , let G (cid:48) = G (cid:116) G and G = G G denote the grid diagrams of their disjoint union and connected sumconstructed using above description. Let x ∈ S ( G ) , x ∈ S ( G ) and ( x ∪ x ) ∈ S ( G G ) . Then A ( x ∪ x ) = A ( x ) + A ( x ) .Proof. First, given the above construction, perform a torus translation on G (cid:48) so that G is positioned on the upper-left and G is positioned in the lower-right corner. Since thereare no new X or O -markings, or intersection points of grid states to the north-east (NE)of the diagrams G and G when forming the disjoint union, there will be no shift in thegradings of x and x viewed as a set of coordinates in G (cid:48) . After interchanging the O -markings and obtaining G , we observe that X -markings and the position of grid states donot change. Hence J ( x ∪ x , x ∪ x ) and J ( X , X ) does not change calculated in G or G (cid:48) .The only additional contribution comes from the relative position of O -markings, whichyields J ( O (cid:48) , O (cid:48) )+1 = J ( O , O ) . This immediately tells us M O ( x ∪ x ) = M O (cid:48) ( x ∪ x )+1 .Notice that G (cid:48) represents a link, so we use the following normalized Alexander functionfor G (cid:48) which takes the number of components into account; A ( x ) = 12 ( M O ( x ) − M X ( x )) − ( n − l COMBINATORIAL DESCRIPTION OF THE KNOT CONCORDANCE INVARIANT EPSILON 9
In this case l = 2 and combining this with the Maslov grading shift above yields theresult. (cid:3) To show that ε defined using grid diagrams is a concordance invariant, we need toprove some properties of ε , analogous to that of ε in knot Floer homology. For thatpurpose, we will be using the following results from [13]. Proposition 3.2 ([13] Proposition 2.6.11) . Suppose that two knots K , K can be con-nected by a smooth, oriented, genus g cobordism W . Let U b ( K ) and U d ( K ) denotethe disjoint unions of K and K with b and d -many unlinked, unknotted componentsrespectively. Then there are knots K (cid:48) , K (cid:48) and integers b, d with the following properties: • U b ( K ) can be obtained from K (cid:48) by b simultaneous saddle moves. • K (cid:48) and K (cid:48) can be connected by a sequence of g saddle moves. • U d ( K ) can be obtained from K (cid:48) by d simultaneous saddle moves. Lemma 3.3 ([13] Lemma 8.4.2) . Let L be an oriented link, and let L (cid:48) = U ( L ) . Thenthere is an isomorphism of bigraded F [ U ] -module cGH − ( L (cid:48) ) (cid:39) cGH − ( L )[[1 , (cid:77) cGH − ( L ) Proposition 3.4 ([13] Proposition 8.3.1) . Let W = F (0 , ⊕ F ( − , − . If L (cid:48) is obtainedfrom L by a split move, then there are F [ U ] -module maps σ : cGH − ( L ) ⊗ W → cGH − ( L (cid:48) ) µ : cGH − ( L (cid:48) ) ⊗ W → cGH − ( L ) with the following properties • σ is homogenous of degree ( − , , • µ is homogenous of degree ( − , − , • µ ◦ σ is multiplication by U , • σ ◦ µ is multiplication by U . We can now prove Theorem 1.1.
Proof of Theorem 1.1.
First, we argue for part a) of Theorem 1.1. By Lemma 3.3, cGH − ( U b ( K )) /T ors (cid:39) ( cGH − ( K ) /T ors ) b . Recall that cGH − ( K ) /T ors (cid:39) F [ U ] . Let α be the generator of the F [ U ] -module.Let G be a diagram for the knot K . Now if K is slice, − K is also slice, i.e. both K and − K are concordant to the unknot. By Proposition 3.2, there exists K (cid:48) such that U b ( K ) can be obtained from K (cid:48) by b simultaneous saddle moves. We arrange the saddlemoves such that we perform them one at a time. Now by the Proposition 3.4 and bythe injectivity of the maps σ and µ , the homology class represented by α is non-zeroin cGH − ( U b ( − K )) /T ors . In this case, we can observe that the grid state x τ in S ( G ) representing the homology class of α maps to a non-torsion homology class in GH − ( − G ) .This is the same homology class represented by the grid state that we obtain by reflecting G and interchanging X and O markings. This can be seen as follows: starting with agrid diagram of K , one can obtain a grid diagram representing the unknot through themoves described in Proposition 3.2. Reflect this unknot diagram and interchange X , O markings and we get the grid diagram of the orientation reversed unknot. Following this, we repeat the moves listed above from K to unknot but now in the reverse order,to obtain the grid diagram − G of − K (this can be done since − K is also concordantto unknot). In every step, α is mapped to a non-torsion element in the homology of theresulting grid diagram and also [ x τ ] is mapped to the same element that is obtained byreflecting G and interchanging X , O markings.Hence, α is a non-zero element in cGH − ( − K ) /T ors . This implies that ε ( K ) = 0 To prove part b), we rely on the symmetry properties of the grid homology and thedual complex (which is the grid complex for − K ) constructed in [13] (Proposition 7.4.3).The explicit isomorphism in this proposition maps all grid states of G to grid states in (cid:57) G . In particular, the non-torsion part of GH − ( G ) is supported at bigrading ( − τ, − τ ) and the non-torsion part of GH − ( (cid:57) G ) is supported at (2 τ, τ ) . As a result, we have GH − ( (cid:57) G ) (cid:39) GH − ( G ) ∗ which implies that ε ( (cid:57) G ) = (cid:57) ε ( G ) . (cid:3) To prove c) and d) in Theorem 1.1, first we prove the following lemma.
Lemma 3.5.
Let L , L be two knots in S and [ x τ i ] be the generators of cGH − ( L i ) /Tors (cid:39) F [ U ] for i = 1 , . Then [ x τ ∪ x τ ] is a non-zero element in cGH − ( L L ) /T ors . Before starting the proof, we describe the union ‘ ∪ ’ notation here. Let x and y betwo chains in GC − ( G ) and GC − ( G ) respectively, and assume that we can write themas x = x + x and y = y + y where x i , y i are grid states in G , G . Then the unionnotation is used as follow; x ∪ y = x ∪ y + x ∪ y + x ∪ y + x ∪ y where the union in each one of the summands is as described in Lemma 3.1.First, we observe that if α is the generator of the F [ U ] module GH − ( L L ) /T ors ,then we can argue that [ x τ ∪ x τ ] = p ( U ) · α , for some non-zero polynomial p ( U ) ∈ F [ U ] .Assuming the additivity of τ invariant under connected sum operations (from literaturein knot Floer homology, see [11, Section 7]) i.e. τ ( K K ) = τ ( K ) + τ ( K ) and Lemma 3.1, we see that p ( U ) = 1 and [ x τ ∪ x τ ] is indeed a generator of the non-torsion part of GH − ( L L ) . Hence, to determine the ε of L L , we need to track theelement [ x τ ∪ x τ ] after reflecting the grid diagram of L L with respect to a horizontalaxis and switching all X and O markings.Notice that proving Lemma 3.5 for L (cid:116) L and combining it with Proposition 3.4, theresult will follow for L L . Proof of Lemma 3.5.
First we show that the element [ x τ ∪ x τ ] is non-zero in the ho-mology of GC − ( G (cid:48) ) , where G (cid:48) denotes the grid diagram of L (cid:116) L .Observe that for a grid state x ∈ S ( G (cid:48) ) which has intersection points lying entirelyin the green (upper-right) and the red portion (lower-left) (See Figure [6]), grid statesin ∂ − ( x ) can be of two kinds. In the first case, these grid states in the boundary havecomponents only in the green or the red portion of the grid diagram which means thatthe rectangles between these grid states are rectangles coming from G and G prior to COMBINATORIAL DESCRIPTION OF THE KNOT CONCORDANCE INVARIANT EPSILON 11
XX OOX XNESW SENW
Figure 6.
Rectangular regions marked by dashed lines with vertices y and x (cid:48) X XNW NESW SEOOO OX X
Figure 7. x (cid:48) , y and grid representatives of y when x (cid:48) and y are connectedvia shaded rectanglesforming disjoint union. If ∂ − ( x ) consists of only these type of grid states, then we haveno new contribution in the homology. Hence, the triviality/non-triviality of x in G (cid:48) isdetermined by the individual rectangles counted in G and G by ∂ − and ∂ − (where ∂ − i denote the boundary map in G i ). In the second case, they can differ from x in twointersection points, one of which lie in NW and other one lie in SE grid blocks. Figure[6] and Figure [7] show such examples.Non-triviality of [ x τ ] and [ x τ ] in GH − ( L ) and GH − ( L ) , respectively, combined withthe following construction will give us the necessary result. We start with performing one X : N E and one X : SW stabilization in both G and G at the upper-right and lower-left part of the diagrams. New diagrams contain two extra intersection points comingfrom the intersection points of the newly introduced vertical and horizontal lines. Aftera stabilization move, this new intersection point is added to all grid states which createsa bijection between the grid states before the stabilization and the grid states after thestabilization. As a result, these two new intersection points are added to x τ ∪ x τ afterstabilizing the diagrams as shown in Figure 6.Now we start our proof for the non-triviality of [ x τ ∪ x τ ] ∈ cGH − ( G (cid:48) ) . Let x (cid:48) , y ∈ S ( G (cid:48) ) such that x (cid:48) and y differ in two components in NE and SW part of the grid (ref.Figure [6], where x (cid:48) contains the dark blue dots and y differs from x (cid:48) hollow grey dotsin NW and SE part of the grid, and they agree in all other coordinates). Assume that x (cid:48) is a grid state in the chain x τ ∪ x τ .As discussed earlier, given a grid state x ∈ S ( G ) and another grid state x (cid:48) ∈ ∂ − ( x ) ,then any empty rectangle connecting x and x (cid:48) can be viewed as an empty rectanglein G (cid:48) with the same O -multiplicity by simply taking the union of both grid states withanother grid state x ∈ S ( G ) . Using this idea, we decompose the domains in G (cid:48) intosmaller domains in G and G by finding corresponding grid representatives in G and G . Hence, let x be a grid state in the cycle that represents the homology class [ x τ ∪ x τ ] and let y be another grid state such that there is a rectangle connecting them. We findthe grid representatives of y in G and G such that: • y i ∈ S ( G i ) for i = 1 , . • y i differs from x | G i in two coordinates one of which is the newly introducedintersection point that is closer to the middle of the diagram where G , G meet.Notice that this also determines the other coordinate where these grid statesdiffer. • U · x | G i ∈ ∂ − i y i , i = 1 , . • the sum of the O -multiplicities in the empty rectangles connecting x | G , y and x | G , y is 1 more than the O -multiplicities of the empty rectangle connecting x , y in G (cid:48) .In Figure [6], such grid representatives of y are shown. After finding these grid repre-sentatives, the above points imply; x ∈ ∂ − y ⇒ x | G i ∈ ∂ − i y i , i = 1 , Since x | G i are grid states in the chains x τ i and x τ i / ∈ Im ( ∂ − i ) for i = 1 , , we get that ( x τ ∪ x τ ) / ∈ Im ( ∂ − ) .By construction, ∂ − ( x τ ∪ x τ ) = 0 . This is because if there is some y that differsfrom x (cid:48) in two vertices (where x (cid:48) is a grid state in the chain ( x τ ∪ x τ )) , then both therectangles with those vertices will have X -markings in it and they will not be countedby the differential ∂ − . (see Figure [6]).To complete the proof, we need to show that [ x τ ∪ x τ ] is a non-torsion element in GH − ( G (cid:48) ) . If [ x τ ∪ x τ ] is a torsion element, then for all n > N , U n · [ x τ ∪ x τ ] = 0 ,for some natural number N . This implies that there exists y (cid:48) ∈ GC − ( G (cid:48) ) such that ∂ − y (cid:48) = U n · ( x τ ∪ x τ ) . If this was the case, we can follow the same strategy as beforeand decompose the domains into G and G , for which appropriate O -multiplicities add COMBINATORIAL DESCRIPTION OF THE KNOT CONCORDANCE INVARIANT EPSILON 13 up to produce U n . Using the fact that [ x τ i ] is non-torsion in GH − ( G i ) for i = 1 , , theprevious argument completes the proof. (cid:3) Now, we are ready to prove the behavior of ε under connected sum operation. Lemma 3.6.
Let G and G be grid diagrams for the knots K and K respectively. If ε ( G ) = ε ( G ) , then ε ( G G ) = ε ( G ) and if ε ( G ) = 0 , then ε ( G G ) = ε ( G ) .Proof. First, let ε ( G ) = ε ( G ) = 0 and [ x τ i ] be the generator of GH − ( G i ) /T ors for i = 1 , . Then by definition, we know that [ x τ i ] is non-zero in GH − ( G i ) /T ors and in GH − ( (cid:57) G i ) /T ors i = 1 , . As a result, by Lemma 3.5, [ x τ ∪ x τ ] is non-zero in both GH − ( G G ) and cGH − ( (cid:57) ( G G )) . Hence, ε ( G G ) = 0 .Now assume ε ( G ) = ε ( G ) = 1 . By the definition of ε , we have ∂ − (cid:57) G i ([ x τ i ]) (cid:54) = 0 for i = 1 , . To compute ε ( G G ) , we look at [ x τ ∪ x τ ] in the grid complex of G G after reflecting the diagram and changing X and O markings as before. Observe that inthis complex, we have the following relation; ( ∂ − ([ x τ ]) ∪ [ x τ ]) + ( x τ ∪ ∂ − ([ x τ ])) ∈ ∂ − ([ x τ ∪ x τ ]) which means that we see a non-trivial contribution to the boundary map and ∂ − [ x τ ∪ x τ ] (cid:54) = 0 . Hence, we conclude that ε ( G G ) = 1 . In the case ε ( G ) = ε ( G ) = − , weapply the same argument but start with the diagrams of − K and − K .The cases ε ( G ) = ± and ε ( G ) = 0 follows from the same technique. In short, wedecompose the domain in G G to corresponding the grid representatives. One cansee that the contribution of the domain coming from G determines ε in this case andthis completes the proof. (cid:3) Some computations for Torus Knots and iterated Cables of torusknots
In this section, we compute ε for negative torus knots and examine the behavior of ε under cabling. Proof of Theorem 1.2.
Before we look at the different cases, it is important to note thatin Section 6.3 of [13], it is shown that, the element denoted as x + which takes theintersection points of upper-right corners of each X -marking, is a non-torsion cycle withmaximal Alexander grading when we consider ( − p, q ) torus knots. In other words, thisis the generator that gives the τ of ( − p, q ) torus knots. Hence, we will keep track of thiselement when we carry on the computations for ε ( G − p,q ) .First, we would like to show that ε ( G p,q ) = 0 if q = 1 . In this case, grid index is p + 1 and grid diagram is encoded with following permutations σ O = (1 , , , . . . , p + 1) and σ X = ( p + 1 , , . . . , p ) . Figure [8] shows a schematic picture of this case.As we can see in the diagram, such a diagram will always consist of O -markings onthe diagonal followed by X -markings in the diagonal below with a single X -marking atthe top-left corner. Now if we reflect this diagram with respect to a horizontal axis andchange all X and O -markings, we will obtain a diagram as in Figure [9]; Figure 8.
A schematic diagram for the case q = 1 and black dots repre-sent x + Figure 9.
Grid diagram for (cid:57) G − p, By looking at the diagram of (cid:57) G − p, , we can immediately see that the element weobtain is still x + and independent of the diagram, this element is always a cycle, re-ferred as "canonical grid state" in [13]. Hence, the element we obtain is non-trivial in GH − ( (cid:57) G − p, ) which implies that ε ( G p, ) = 0 .Now assume that q > . We would like to show that ε ( G − p,q ) = − . We will keeptrack of the same element as in the previous discussion, but the diagram will be slightlydifferent. Notice that starting with x + , after reflecting the diagram and changing themarkings, this element becomes o + which denotes the grid state that consists of thelower-right intersection points of O -markings. In this case, we see that for any othergrid state y ∈ S ( (cid:57) G − p,q ) such that o + and y differ in two consecutive coordinates, thereexists exactly one × rectangle with an O-marking that goes from y to o + , and one cansee that for such a grid state y , there are no other elements in the boundary due to theposition of the intersection points, either the rectangles cancel in pairs or they containX-markings (see the Figure [10]). Hence, this element is in the image of the boundarymap ∂ −− G which implies that ε ( G − p,q ) = − for q > . COMBINATORIAL DESCRIPTION OF THE KNOT CONCORDANCE INVARIANT EPSILON 15
Figure 10.
An illustration for G − , , on the right black dots are theelements we keep track of, and small squares represent another elementthat is in the boundary of the distinguished element, and these grid statesagree on all other coordinates. Shaded × square shows an example ofsuch a U contribution to the boundary mapFinal case is when q < , however we appeal to the fact that G p, − q is same as (notconsidering the orientation reversal) (cid:57) G p,q . Hence, ε ( G p,q ) = 1 for q < , following fromthe previous case and part b) of Theorem 1.1. (cid:3) As an example for the case when q > , we consider Figure [11] which is a grid diagramfor the left-hand trefoil or T − , . For this case, consider the grid state on the right, whichdiffers from the distinguished grid state, after mirroring, in the red intersection points.One can see that distinguished grid state multiplied by U is the boundary of that gridstate and there are no other elements in the boundary of the red grid state. Hence, byour definition of ε , ε ( − T , ) = − . (cid:31) (cid:30) (cid:31)(cid:31) (cid:31)(cid:31) (cid:30)(cid:30) (cid:30) (cid:30) (cid:30) (cid:30) (cid:30)(cid:30) (cid:30) (cid:31) (cid:31)(cid:31)(cid:31) (cid:31) (cid:30) (cid:30)(cid:30)(cid:30)(cid:30) (cid:31)(cid:31)(cid:31) (cid:31) (cid:31) (cid:30) (cid:31) Figure 11.
Grid diagram for LHT. The grid state identified by the blackdots in the left most diagram is the generator of the free part of the fullyblocked grid homology for left-hand trefoil. The middle diagram is themirror of the left one and the right-most diagram is after changing themarkings.We would like to continue this discussion by proving the behaviour of ε under ( p, q ) -cabling of negative torus knots. Proof. (Proof of Theorem 1.3)
Before looking at the different cases of ε and how it behaves under cables, we wouldlike to describe here how we obtain ( r, rn − -cable of a negative torus knot usinggrid diagrams, where n is the writhe of G − p,q and r is the cabling coefficient. Afterunderstanding this, we follow a similar path as in [6] and refer to Van Cott’s work [17]to generalize it to ( p, q ) -cables. Throughout the proof, we use G to denote the standardgrid diagram for the negative torus knot T − p,q . We begin by explaining how we obtainthe ( r, rn − from a grid.In any grid diagram G , an X or O -marking is a corner and it can be one of the fourtypes; NW (northwest), NE (northeast), SW (southwest) or SE (southeast). If an O -marking is a SE corner, we use the notation o SE , and a similar notation for other cornersand markings. In the specific case that we have the standard grid diagram for a negativetorus knot G − p,q , we have exactly p -many o SW , p -many o NE , ( q − p ) -many o SE corners.Similarly, we have p -many x SE and q -many x NW corners. To obtain a ( r, rn − -cable,we replace empty squares with r × r empty blocks and for marked squares, we replacethem with r × r blocks with the same marking, also repeating the same corner of themarking. This will give us r parallel copies of G − p,q . To introduce one full negativetwist, we choose the o SE r × r block and introduce the twist as shown in Figure [12] andFigure [13]. The resulting diagram will have r n + ( r − negative crossings where n isthe writhe of the original knot As pointed out in [3], this is the total number of crossingswe should have after cabling. Figure 12.
A local picture on a grid diagram where r = 3 and we repeatall the marked corners with respect to their directionsa) Assume now ε ( G ) = 0 . By Theorem 1.2, this is only possible when | q | = 1 and this is the case when we have the unknot. Hence, if we apply the abovecabling procedure, we will obtain a torus knot. This immediately tells us that COMBINATORIAL DESCRIPTION OF THE KNOT CONCORDANCE INVARIANT EPSILON 17
Figure 13.
We introduce a full negative twist by changing the O -markings in o SE corner. This operation produces ( r − -many new nega-tive crossings. ε ( K r,rn − ) = ε ( G r,rn − ) where r is the cabling coefficient and n is the writhe ofthe original knot.b) More interesting case is when ε ( G ) (cid:54) = 0 . In this case, we start with a non-trivialnegative torus knot. Assume that q > , so that ε ( G ) = − . Recall that, asdiscussed in the proof of Theorem 1.2, the distinguished cycle x + is the elementwe would like to keep track of. After applying the above cabling operation, x + in the resulting diagram still has the maximal Alexander grading and it is acycle. Using Proposition 6.4.8 in [13] we get that this element is non-torsion aswell. Hence, we would like to keep track of this element to compute the valueof epsilon. Following our description, we reflect this diagram with respect to ahorizontal axis and change X and O -markings. As before, this grid state is nowin the lower-right corners of O -markings that we denoted as o + . Now we trackthis distinguished element in the grid diagram of the ( r, +1) cable of n -framed T − p,q to find ε (( T − p,q ) r,rn +1 ) .Observe that we can find the ( r, -cable of an n -framed T − p,q , reflect thatand interchange X , O -markings or equivalently, we can find ( r, -cable of an n -framed T p,q . Moreover we can perform the twist in the cable at an o SW corner,so that obtain a ( r, rn + 1) -cable. Now we see that around each of the r -clustersof O -markings, we can perform successive grid commutation moves so that the r -clusters of O -marking lie anti-diagonally (see Figure [14] and Figure [15]). Itis now easy to see that the boundary of o + is zero and we can find another grid state the boundary of which contains only U · o + , see the picture on the right sideof the Figure [15]. The boundary of the element only contains U · o + since theboundary of o + is zero. As a result, ε (( T − p,q ) r,rn +1 = − ε ( T − p.q ) . The casewhen q < follows from a similar argument and completes the proof. (cid:3) Remark 4.1.
It is important to note here that a similar technique can be appliedfor iterated cables of torus knots, since in the grid diagram obtained after cabling theoriginal torus knot, x + still gives us the non-zero, non-torsion element with the ‘correct’Alexander grading i.e. one can track x + (and thus o + ) to find ε of the iterated cableand the argument for those cases hold similarly. XXXXXX
OOO
X XXXXX
O OO
Figure 14.
Commutation moves along one of non-twisting O -markingcluster XO X X XXX
O O
X XXX X XOO O
Figure 15.
Commutation moves along the twisting O -marking clusterand a grid state killing U · o + COMBINATORIAL DESCRIPTION OF THE KNOT CONCORDANCE INVARIANT EPSILON 19 Grids and Braids
In this section, we recall the relation between grids and braids. We follow the notationconvention of Ng and Thurston from [9].Given a grid diagram G , we can view it as a braid. We still require that vertical strandsgo over horizontal strands. We also add the condition that all horizontal segments areoriented from left to right on the diagram. To achieve this, given a grid diagram G , if an O marking on a row is to the left of an X marking, join them as usual inside the grid.If an O marking is to the left of an X marking, then draw a line segment that pointsrightwards from the O marking and another line segment that connect to the X markingon the same row through the identified edged on the toroidal diagram. By the aboveconstruction, if G is a grid diagram for a knot K , then braid closure of B is K . Thisconstruction produces a rectilinear braid diagram and after perturbing it accordingly,we obtain a braid diagram. An example is given in Figure 16. Figure 16.
A braid in B with braid word σ σ σ σ . This is also anexample of a positive braid.Any braid in B n can be written as a product of the standard generators σ (cid:15) i i for i =1 , . . . , n − and (cid:15) i ∈ {± } . In our convention, we enumerate the strands from top tobottom and a generator σ i refers to i th strand going over ( i + 1) st strand on the diagram.Notice also with this convention, (cid:15) i agrees with the sign of the crossing.Now, we turn our focus to a particular family of braids, namely positive braids. Abraid B ∈ B n is called a positive braid if in its braid word all (cid:15) i are positive. This is alsoequivalent to having all positive crossings. In [16], Stallings proves a more general resultfor homogeneous braids, which directly implies that positive braids are fibered. Com-bining this result with Hedden’s result about strongly quasipositive knots (Proposition2.1) in [4], we can conclude that the canonical grid state denoted as x + is the elementthat computes τ and therefore, this is the element we need to keep track of when wecompute ε for positive braids. Proof of Theorem 1.4.
Our proof relies on the observation that if a grid diagram G + represents a positive braid, then we are limited to very particular local pictures and arelative X and O -marking patterns. First of all, recall that with this convention, sign ofthe markings agree with (cid:15) i ’s that appear in the braid word. Hence, we only have positivecrossings on the grid diagram. Since all the horizontal strands are oriented from left toright and from O to X , we have the local pictures as shown in Figure 17. Figure 17.
Different schematic local pictures of possible positive cross-ings. Black dots are intersection points from x + Notice that the local pictures shown in Figure 17 represent all possible positive crossingeither they occur in the middle of the grid diagram (1), or they occur on horizontalstrands that are joined outside the grid diagram (2) and (3). We mark the intersectionpoints of x + with black dots. We reflect the grid G + with respect to a horizontal axis andchange the markings. Similar to previous computations, the distinguished intersectionpoints are now in the lower-right corners of O -markings. After this, we perform onelocal modification that allows us to compute epsilon. At the O -marking that is locatedat the bottom of the vertical strand, we apply an O : N W type stabilization. Asstated in Proposition 5.2.1 in [13], this does not change the homology, so we keep trackof this distinguished element in the same homology after the stabilization. Moreover,stabilization creates a new pair of α and β curve and their new intersection point isautomatically added to all grid states, including o + . This is a one-to-one correspondencebetween grid states, hence we still use the same notation for this new grid state. Thereare exactly two rectangles from the grid state x + to y , one is the shaded rectangle inthe figure and the other rectangle wraps around the torus in the complement. One cansee that the shaded rectangle has non-trivial U contribution whereas the other rectanglecontain an X -marking. This implies that y ∈ ∂ −− G ( x + ) and as a result, [ x + ] / ∈ Ker ( ∂ −− G ) . The same strategy also proves the existence of such a rectangle with non-trivial U contribution for other local positive crossing pictures. Hence, we conclude ε ( G + ) = 1 . (cid:3) As a straightforward corollary of Theorem 1.4, we have the following statement;
Corollary 5.1.
Let K be a positive knot that can be realized as the braid closure of apositive braid, and let G be a grid diagram for K , then ε ( G ) = 1 Clearly, positive knots form a larger family than positive braids. Proving a similarresult for positive knots has the difficulty that we cannot guarantee such a local picture
COMBINATORIAL DESCRIPTION OF THE KNOT CONCORDANCE INVARIANT EPSILON 21
Figure 18. O : N W stabilization performed on the grid in case (1). Thegrid state y is marked with small squares and the gray shaded area is thedomain with non-zero contributionas in Figure [17], and as a result, the same problem becomes harder to tackle using griddiagrams. However, the corollary above directly follows from Theorem 1.4. References [1] Kenneth L. Baker, J. Elisenda Grigsby, and Matthew Hedden. Grid diagrams for lens spaces andcombinatorial knot Floer homology.
Int. Math. Res. Not. IMRN , 2008.[2] John A. Baldwin and William D. Gillam. Computations of Heegaard-Floer knot homology.
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Grid homology for knots and links , volume208. American Mathematical Soc., 2015.[14] Peter S. Ozsváth, András I. Stipsicz, and Zoltán Szabó. Concordance homomorphisms from knotFloer homology.
Adv. Math. , 315, 2017.[15] Sucharit Sarkar and Jiajun Wang. An algorithm for computing some Heegaard Floer homologies.
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Department of Mathematics, University at Buffalo
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