Kauffman states and Heegaard diagrams for tangles

Abstract

We define polynomial tangle invariants ∇ s T via Kauffman states and Alexander codes and investigate some of their properties. In particular, we prove symmetry relations for ∇ s T of 4-ended tangles and deduce that the multivariable Alexander polynomial is invariant under Conway mutation. The invariants ∇ s T can be interpreted naturally via Heegaard diagrams for tangles. This leads to a categorified version of ∇ s T : a Heegaard Floer homology HFT for tangles, which we define as a bordered sutured invariant. We discuss a bigrading on HFT and prove symmetry relations for HFT of 4-ended tangles that echo those for ∇ s T .Kauffman states and Heegaard diagrams for tangles 3 of L and we call R the mutating tangle in this mutation. If L is oriented, we choose an orientation of L that agrees with the one for L outside of R. If this means that we need to reverse the orientation of the two open components of R, then we also reverse the orientation of all other components of R during the mutation; otherwise we do not change any orientation. For an alternative, but equivalent definition, see definition 3.3 and remark 3.4.Theorem 0.2 along with the glueing formula for ∇ s T gives rise to the following result. Corollary 0.4 (3.5) The multivariate Alexander polynomial is invariant under Conway mutation after identifying the variables corresponding to the two open strands of the mutating tangle.This result has long been known for the univariate Alexander polynomial, see for example [LM87, proposition 11], but I have been unable to find a corresponding result for the multivariate polynomial in the literature. The fact that mutation invariance follows so easily from the symmetry relations of theorem 0.2 suggests that ∇ T is well-suited for studying the "local behaviour" of the Alexander polynomial. The homological tangle invariant HFTHeegaard Floer homology theories were first defined by Ozsváth and Szabó in 2001 [OS01]. With an oriented, closed 3-dimensional manifold M , they associated a family of homological invariants, the simplest of which is denoted by HF(M). Given an oriented (null-homologous) knot or link L in M , Ozsváth and Szabó, and independently J. Rasmussen, then defined filtrations on the chain complexes which give rise to the respective flavours of knot and link Floer homology [OS03a, Ras03, OS05], the simplest of which is denoted by HFL(L). The Alexander polynomial can be recovered from these groups as the graded Euler characteristic.Given corollary 0.4, it is only natural to ask for a Heegaard-Floer theoretic categorification of ∇ s T . To this end, we define a homology theory HFT as follows: given a Heegaard diagram for a tangle T (see definition 4.1) along with a site s of T , we define a finitely generated Abelian group which comes with two gradings: a relative homological Z-grading and an Alexander grading, which is an additional relative Z-grading for each component of the tangle:CFT(T, s) = h∈Z ←homological grading a∈Z |T| ←Alexander grading CFT h (T, s, a).

Peculiar modules for 4‐ended tangles

“…Let us start by recalling the basic definitions from [41,Sections 1,4], adapted to 4-ended tangles. tangle end and following the orientation of the fixed circle S 1 , we number the tangle ends and label the arcs S 1 im(T ) by a, b, c and d, in that order.…”

“…Obviously, the Heegaard moves from [41,Lemma 4.13] are equivalent to the following moves for peculiar Heegaard diagrams:…”

“…Let us briefly recall the definition of the Heegaard Floer homology HFT(T ) from [41,Section 5], adapted to 4-ended tangles.…”

“…When comparing this to [41,Definition 5.13], note that we now use peculiar Heegaard diagrams to compute the Maslov index μ. Furthermore, for every component t of the tangle, there is a relative Z-grading A t , which is called the Alexander grading, see [41,Definition 5.6]. Given an ordered matching…”

“…We sometimes denote the Alexander grading on generators by a superscript list of integers (or half-integers, if, for example, we want to achieve the same symmetry present in the decategorified invariants from [41]), like a +1 for the univariate or a ( 3 2 ,− 1 2 ) for the multivariate grading.…”