Peculiar modules for 4‐ended tangles

Abstract

With a 4‐ended tangle T, we associate a Heegaard Floer invariant prefixCFT∂false(Tfalse), the peculiar module of T. Based on Zarev's bordered sutured Heegaard Floer theory (Zarev, PhD Thesis, Columbia University, 2011), we prove a glueing formula for this invariant which recovers link Floer homology HFL̂. Moreover, we classify peculiar modules in terms of immersed curves on the 4‐punctured sphere. In fact, based on an algorithm of Hanselman, Rasmussen and Watson (Preprint, 2016, arXiv:1604.03466v2), we prove general classification results for the category of curved complexes over a marked surface with arc system. This allows us to reinterpret the glueing formula for peculiar modules in terms of Lagrangian intersection Floer theory on the 4‐punctured sphere. We then study some applications: Firstly, we show that peculiar modules detect rational tangles. Secondly, we give short proofs of various skein exact triangles. Thirdly, we compute the peculiar modules of the 2‐stranded pretzel tangles T2n,−false(2m+1false) for n,m>0 using nice diagrams. We then observe that these peculiar modules enjoy certain symmetries which imply that mutation of the tangles T2n,−false(2m+1false) preserves δ‐graded, and for some orientations even bigraded link Floer homology.

Knot Floer homology of some even 3-stranded pretzel knots

“…where µ(s, δ) = s−δ is the Maslov grading corresponding to Alexander grading s and delta grading δ. Conversely, we see that if, for a given Alexander grading s, HF K(K, s) is supported in a single delta grading, then the rank of Our computation of the Knot Floer homology will be accomplished using a Heegaard Floer invariant for 4-ended tangles developed by Zibrowius [8]. For our purposes, we shall be interested in the invariant HF T (T ) which, for an oriented 4-ended tangle T , consists of a collection of (graded) immersed curves on the parametrized 4-punctured sphere S 2 4 .…”

“…Theorem 2.1 (Theorem 5.9 of [8]). Let T 1 and T 2 be (appropriately oriented) 4-ended tangles, and suppose that their pairing results in a knot K. Then HF K(K) ⊗ V ∼ = HF (HF T (mr(T 1 ), HF T (T 2 ))…”

“…We wish to give a description of the tangle invariant HF T for the (2a, −2b − 1)-pretzel tangle (pictured in Figure 8). First, we shall recall the two types of standard curve invariants from [8]. These are the rational and special curves.…”

“…The rest of this work is organized as follows: in the next section, we give a brief outline of Knot Floer homology and the tangle invariant HF T introduced by Zibrowius [8], and we describe how to obtain Knot Floer homology by pairing tangle invariants. We conclude that section by enumerating the different tangle invariants that will arise in our computations for the pretzel knots we are interested in.…”

The Fukaya category of the pillowcase, traceless character varieties, and Khovanov cohomology

“…Originally defined using the symplectic field theoretic framework of Lipshitz's cylindrical version of Heegaard Floer theory [26], the bordered theory was recast in terms of A ∞ -modules over a particular A ∞ -algebra generated by a collection of Lagrangians in the partially wrapped Fukaya category of a symmetric product of the boundary surface (suitably punctured) [5,6,24]. Important special cases have been worked out in concrete detail; for 3manifolds with torus boundary by Hanselman-Rasmussen-Watson [12], and for 2-stranded tangles by Zibrowius [10]. These are formally analogous to our framework, and our invariant should be viewed as a type of bordered Khovanov invariant for a 2-tangle.…”