Skein relations for tangle Floer homology

Skein relations for tangle Floer homology

Ina Petkova Department of Mathematics, Dartmouth College, Hanover, NH 03755, USA ina.petkova@dartmouth.edu http://www.math.dartmouth.edu/~ina/  and  C.-M. Michael Wong Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, USA cmmwong@lsu.edu http://www.math.lsu.edu/~cmmwong/
Abstract.

In a previous paper, Vértesi and the first author used grid-like Heegaard diagrams to define tangle Floer homology, which associates to a tangle a differential graded bimodule . If is obtained by gluing together , then the knot Floer homology of can be recovered from . In the present paper, we prove combinatorially that tangle Floer homology satisfies unoriented and oriented skein relations, generalizing the skein exact triangles for knot Floer homology.

Key words and phrases:
tangles, knot Floer homology, bordered Floer homology, skein relation
2010 Mathematics Subject Classification:
57M58 (Primary); 57M25, 57M27 (Secondary)

1. Introduction

Heegaard Floer homology is an invariant of closed, oriented -manifolds introduced in [29] that has found numerous applications in recent years, and is known to be equivalent [16, 17, 18, 19, 20] to monopole Floer homology [15], and also equivalent [4] to embedded contact homology [6, 7, 8]. In [28, 36], it is extended to give an invariant, knot Floer homology, of null-homologous knots in closed, oriented -manifolds, which is further generalized to oriented links in [32]. Knot Floer homology comes in many flavors; its simplest form, for an oriented link , is a bigraded module over or . There is a combinatorial description of the knot Floer homology of links called grid homology [26, 27, 33], defined using grid diagrams. Because knot Floer homology, defined analytically, is known to categorify the Alexander polynomial, it is often compared with Khovanov homology [12, 13], a link invariant from representation theory that categorifies the Jones polynomial.

Ozsváth and Szabó [30] show that the Heegaard Floer homology of the branched double cover of a link satisfies an unoriented skein exact triangle, from which they derive a spectral sequence from to , thus relating the two theories. Following this, Manolescu [24], by counting holomorphic polygons, shows that knot Floer homology with coefficients also satisfies an unoriented skein exact triangle, and uses it to show that for quasi-alternating links. Manolescu and Ozsváth [25] then use the skein exact sequence to show that quasi-alternating links are Floer-homologically -thin over .

While the discussion above seems to suggest that there may be a spectral sequence relating and that comes from iterating Manolescu’s skein relation, Baldwin and Levine [1] discover that the page of the spectral sequence they so construct is not even a link invariant. However, one may be able to relate the two theories with some modifications: Baldwin, Levine, and Sarkar [2] construct another spectral sequence that converges to for some module of rank and some integer , where the differential counts some of the holomorphic polygons in Manolescu’s unoriented skein sequence. They conjecture that the page of this spectral sequence coincides with a variant of Khovanov homology for pointed links, the proof of which would imply a version of the following conjecture, first formulated by Rasmussen [35] for knots:

Conjecture 1.

For any -component link , we have

To better understand Manolescu’s skein relation and the related conjectures, there are several approaches. One idea involves computing the maps in the skein relation combinatorially: The second author [38] gives a version of the skein sequence for grid homology, generalizing the results on quasi-alternating links [24, 25] to -coefficients and giving a spectral sequence from a cube-of-resolutions complex with no diagonal maps. Lambert-Cole [21] exploits the computability in [38] to show that -graded knot Floer homology is invariant under Conway mutation by a large class of tangles.

Another idea is to understand the maps in the skein relation on a local level, by slicing the links involved into tangles and studying a tangle version of knot Floer homology. One such theory is tangle Floer homology, defined by Vértesi and the first author [34]. In this theory, to a sequence of points one associates a differential graded algebra, and to a tangle one associates an -module over the differential graded algebra(s) associated to its boundary. If a link is obtained by gluing together tangles , then can be recovered by taking a suitable notion of tensor product, called the box tensor product, of . The -modules are defined combinatorially using nice diagrams (in the sense of Sarkar and Wang [37]) that are similar to grid diagrams. The tangle Floer package is inspired by bordered Floer homology, an invariant of -manifolds with parametrized boundary that can be used to recover the Heegaard Floer homology of a manifold obtained by gluing, defined by Lipshitz, Ozsváth, and Thurston [22].

Similar to knot Floer homology, tangle Floer homology also comes in multiple flavors. For example, is an ungraded type  structure, is a -graded type  structure, and is an -bigraded type  structure, where and are the Maslov and Alexander gradings respectively. As the notation suggests, these structures do not depend on the choice of Heegaard diagram for , but only on the number of markers in , which we denote by . There is also a richer bigraded version, , which recovers the richer version of knot Floer homology , and which is believed but not yet proven to be an invariant of . We postpone the precise definitions of these, as well as other type , type , and type  structures, to Section 2.

The first part of this paper addresses the idea above; namely, we prove an unoriented skein relation for tangle Floer homology. Suppose , , and are three unoriented tangles in identical except near a point, as in Figure 1.1.

\labellist\pinlabel

at 25 55 \pinlabel at 152 55 \pinlabel at 280 55 \endlabellist

Figure 1.1. Top: Three tangles , , form an unoriented skein triple if they are identical except near a point, as displayed. Bottom: A specific example of an unoriented skein triple.
Theorem 2.

There exists a type  homomorphism such that

as ungraded type  structures. Analogous statements hold for type , , and structures.

In fact, we prove a strengthened version of Theorem 2 for oriented tangles, taking into account the -grading. Suppose , , and are three tangles as above, but oriented, and choose corresponding oriented planar diagrams that are identical (after forgetting the orientations) except near a point. Let denote the number of negative crossings in the diagram for , and let and .

Theorem 3.

There exists a type  homomorphism of -degree such that

as -graded type  structures. Analogous statements hold for type , , and structures.

Remark.

Following [33, 34], our -gradings differ from those in [25, 38] by a factor of .

By taking the box tensor product, we immediately obtain a combinatorially computable unoriented skein exact triangle for knot Floer homology, recovering a version of the results in [24, 38]. Suppose , , and are three oriented links that are identical (after forgetting the orientations) except near a point, so that they form an unoriented skein triple. Let , , and be the number of components of , , and respectively, and define , , and in a fashion analogous to , , and above.

Corollary 4.

For sufficiently large , there exists a -graded exact triangle

where is a vector space of dimension with grading , and is a vector space of dimension with grading .

Remark.

Due to a difference in the orientation convention, the arrows in the exact triangle point in the opposite direction from those in [24, 25]. We follow the convention in [28, 26, 27, 38], where the Heegaard surface is the oriented boundary of the -handlebody.

In another direction, Theorem 3 may also provide a way to further the development of knot Floer homology in the framework of categorification. Precisely, tangle Floer homology has been shown by Ellis, Vértesi, and the first author [5] to categorify the Reshetikhin–Turaev invariant for the quantum group . This puts tangle Floer homology on a similar footing as the tangle formulation of Khovanov homology [13, 3, 39], which categorifies the Reshetikhin–Turaev invariant for . What is missing in the work of Ellis, Vértesi, and the first author is a construction of -morphisms, corresponding to tangle cobordisms.

For knot Floer homology, cobordism maps are defined by Juhász [9] using contact geometry, and independently by Zemke [40] using elementary cobordisms, and together they [11] show that their definitions coincide. Juhász and Marengon [10] prove that the cobordism maps in [9] fit into a skein exact triangle, providing evidence that these cobordism maps are actually the maps in skein sequences. Thus, one approach to constructing the -morphisms mentioned above is to study the skein relations of tangle Floer homology further.

Continuing in this line of thought, we also prove an oriented skein relation for tangle Floer homology in the second part of this paper, which generalizes an analogous relation for knot Floer homology proven by Ozsv́ath and Szabó [28, 31]. Suppose , , and are three oriented elementary tangles identical except near a point, with the strands at which the tangles differ oriented from right to left, as in Figure 1.2.

Figure 1.2. From left to right, the elementary tangles , , .

There are corresponding Heegaard diagrams , , and , which we describe explicitly in Section 5. Below, and are variables corresponding to the strands at which the tangles differ.

Theorem 5.

There exists a type  homomorphism of -degree such that

as -bigraded type  structures.

Remark.

Since is not yet known to be a tangle invariant [34], Theorem 5 is stated for the type  bimodules of Heegaard diagrams rather than for bimodules associated to a tangle.

Remark.

Tangle Floer homology is currently only defined over , and so the negative signs in Theorem 5 could be replaced by positive signs. However, the stated signs are what one would expect for a theory defined over . This remark applies also to Lemma 5.4 and Lemma 5.10.

Restricting to , we also obtain a local oriented skein relation for that version. In this case, we have a proven tangle invariant, so the relation holds for general tangles. Suppose , , and are three tangles that form an oriented skein triple, as in Figure 1.3.

\labellist\pinlabel

at 25 55 \pinlabel at 152 55 \pinlabel at 280 55 \endlabellist

Figure 1.3. Top: Three tangles , , form an oriented skein triple if they are identical except near a point, as displayed. Bottom: A specific example of an oriented skein triple.
Theorem 6.

There exists a type  homomorphism of -degree such that

as -bigraded type  structures. Analogous statements hold for type , , and structures.

Again by taking the box tensor product, we obtain an oriented skein exact triangle for knot Floer homology, recovering a version of the results in [28, 31]. Suppose , , and are three oriented links that are identical except near a point, so that they form an oriented skein triple.

Corollary 7.

If the two strands of belong to the same component, then there exists an -bigraded exact triangle

and if the two strands of belong to different components, then there exists an -bigraded exact triangle

where is a module of rank with bigradings and , and is a module of rank with bigradings and . Analogous statements hold for .

Since tangle Floer homology shares some similarities with grid homology, our approach to proving Theorems 2 and 3 is similar to that in [38], and our approach to proving Theorem 5 is similar to that in [33, Chapter 9]. In particular, all maps involved are combinatorially computable.

Organization

We review the necessary algebraic background and the definition of tangle Floer homology in Section 2. We prove the ungraded unoriented skein relation, Theorem 2, in Section 3. We then determine the -gradings in Section 4 to prove the graded skein relation, Theorem 3. Theorems 5 and 6 is proven in Section 5.

Acknowledgments

We thank Robert Lipshitz and Vera Vértesi for useful conversations, and we also thank Robert Lipshitz for corrections on an earlier draft. IP received support from an AMS-Simons travel grant and NSF Grant DMS-1711100. IP thanks Louisiana State University, and MW thanks Rice University and Dartmouth College for their hospitality while this research was undertaken.

2. Background

2.1. Algebraic structures

We first review the underlying algebraic structures of tangle Floer homology. We will only define the immediately relevant structures here, and refer the interested reader to [23, Section 2].

Let be a unital differential graded algebra (DGA) with differential and multiplication over a base ring of characteristic . In this paper, will always be the ring of idempotents, which is a direct sum of copies of . We will also write to denote for algebra elements , whenever no confusion can arise.

A (left) type structure over is a graded -module equipped with a homogeneous map

satisfying the compatibility condition

It may be advantageous to represent this graphically:

The map is called the structure map of . Defining

inductively by

we say that is bounded if for all , there exists an integer such that for all .

Let be two unital differential graded algebras, with differentials and , and multiplications and , over the base rings respectively. (Recall that the base rings have characteristic .) A (left-right) type structure over is a graded -bimodule equipped with a homogeneous structure map

satisfying the compatibility condition

Graphically, this can be represented as:

Like for type  structures, we can define and the notion of boundedness analogously. Type  structures are the main objects of study in this paper. We will denote type  structures by calligraphic letters (e.g. ), and reserve for the underlying -bimodules.

A morphism of degree is simply a module homomorphism

(By abuse of notation, we use to denote both maps above.) Given a morphism, we can define its boundary by

or graphically,

For convenience, although this is not found in the literature, we will use the notation to represent the last two terms above:

Given two morphisms of degree and of degree , where , , are type  structures over , their composition , of degree , is defined as the map

given by

or graphically,

Note that the structure map can be thought of as a morphism , and so we can consider the morphisms and also. In this notation, we can write

The above operations make type  structures over a differential graded category.

A type homomorphism (or simply a homomorphism) from to of degree is a morphism satisfying . Graphically, this can be represented as

For example, the identity morphism of a type  structure is the map that sends to , where (resp. ) is the unit of the algebra (resp. ). In the context of tangle Floer homology, and will be the sum of all primitive idempotents.

Given a homomorphism of degree between two type  structures over , the mapping cone of is the type structure with underlying -bimodule and structure map given by

Let be two homomorphisms. A homotopy between and is a morphism such that

or graphically,

Note that is a morphism, but not a homomorphism unless . We write if and are homotopic. We say that two type  structures are homotopy equivalent, and write , if there exist type  homomorphisms and such that is homotopic to and is homotopic to . In the full subcategory of type  structures that are homotopy equivalent to bounded ones, the notion of homotopy equivalence coincides with an appropriate notion of quasi-isomorphism [23, Corollary 2.4.4]. All algebraic structures in bordered Heegaard Floer homology and tangle Floer homology are homotopy equivalent to bounded ones; this can be seen by choosing an admissible Heegaard diagram that defines the same bordered -manifold or tangle [23, Lemma 6.6].

Although we will only focus on type —in fact, type  structures—we should mention that there are also type  structures over a differential graded algebra , which (in the present context) are just differential graded modules over . By extension, there are also type , , and structures. There is a box product (or box tensor) operation between a right (resp. left) type  structure and a left (resp. right) type  structure (at least one of which is bounded), resulting in a chain complex (resp. ) over . The box tensor is defined also for bimodules; for example, box-tensoring a type  structure and a type  structure yields a type  structure . We refer the interested reader to [23, §2.3.2].

We may treat itself as a type  structure; box-tensoring with then turns a type  structure into a type  structure . In fact, this defines a differential graded functor from the full subcategory of type  structures that are homotopy equivalent to bounded ones to the full subcategory of type  structures that are homotopy equivalent to bounded ones. This functor is actually a quasi-equivalence [23, Proposition 2.3.18], implying that it preserves quasi-isomorphisms. Corresponding statements hold for type  and type  structures. Since the notions of quasi-isomorphism and homotopy equivalence coincide for structures of any type given that they are homotopy equivalent to bounded ones [23, Corollary 2.4.4], to prove Theorem 2, we need only prove it for type  structures.

In our proof of Theorem 2, we will need to adapt to the setting of type  structures a lemma in homological algebra, whose version for chain complexes first appeared in [30].

Lemma 2.1.

Let be a collection of type  structures over and , which are both unital differential graded algebras over a base ring of characteristic , and let , , and be morphisms satisfying the following conditions for each :

  1. The morphism is a type  homomorphism, i.e.

  2. The morphism is homotopic to zero via the homotopy , i.e.

  3. The morphism is homotopic to the identity via the homotopy , i.e.

(A graphical representation of the conditions above is given in Figure 2.1.) Then for each , the type  structure is homotopy equivalent to the mapping cone .

(1)
(2)
(3)
Figure 2.1. Graphical representations of the conditions in Lemma 2.1.
Proof.

Observe that the mapping cone is defined because is a type  homomorphism, using Condition (1). It has underlying module , and structure map

To show that , we define the morphisms and as follows:

We first claim that and are type  homomorphisms. Indeed,

where the last equality follows from Conditions (1) and (2). Similarly,

where again the last equality follows from Conditions (1) and (2).

We next claim that and . To show this, we define the homotopy morphisms and as follows:

Then

where the last equality follows from Condition (3). The homotopy is a little more tedious.

where the last equality uses Conditions (2) and (3), and is the morphism

Letting be the morphism

we see that . Observe also that . But then

This shows that , as desired. ∎

Remark.

In [30], a proof is given for the chain-complex version of this lemma. That proof does not translate to the case of type  structures, since it involves taking the homology of the chain complexes. Instead, the proof we have presented here is the type  version of an alternative proof for the lemma in [30, 14, 24, 38], which is known in the community but not found in the literature.

In our proof of Theorem 5, we will also use another lemma in homological algebra, which is the analog of a well-known lemma for chain complexes.

Lemma 2.2.

Let be type  structures over and , which are both unital differential graded algebras over a base ring of characteristic . Let be a homotopy equivalence of type  structures, and for , let be type  homomorphisms such that

Then is homotopy equivalent to .

Proof.

By definition, there is a type  homomorphism such that and for some morphisms . The mapping cone has underlying module , and structure map

We define the morphisms and by

We first claim that and are type  homomorphisms. Indeed,