Combinatorial tangle Floer homology

Abstract

In this paper we extend the idea of bordered Floer homology to knots and links in $S^3$: Using a specific Heegaard diagram, we construct gluable combinatorial invariants of tangles in $S^3$, $D^3$ and $I\times S^2$. The special case of $S^3$ gives back a stabilized version of knot Floer homology.Comment: 106 pages, 44 figure

Skein relations for tangle Floer homology

“…bijections s → t for s, t ⊂ [n]), which can be drawn as strand diagrams (up to isotopy and Reidemeister III moves), as follows. Represent each p i by a horizontal orange strand [0, 1] × {i − 1 2 } oriented left-to-right if p i = + and right-to-left if p i = − (in [34], those are dashed green strands and double orange strands, respectively). Represent a bijection ϕ : s → t by black strands connecting (0, i) to (1, ϕi) for i ∈ s. We further require that there are no triple intersection points and there are a minimal number of intersection points between strands.…”

“…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 T ⊂ I × R 2 one associates an A ∞ -module CT(T ) over the differential graded algebra(s) associated to its boundary.…”

“…Analogous statements hold for type DA, AD, and AA structures.Remark. Following [33,34], our δ-gradings differ from those in [25,38] by a factor of −1.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 L ∞ , L 0 , and L 1 are three oriented links that are identical (after forgetting the orientations) except near a point, so that they form an unoriented skein triple.…”

On the decategorification of Ozsváth and Szabó's bordered theory for knot Floer homology

“…In [PV14], Petkova and Vértesi define a theory with similar properties to Ozsváth and Szabó's, which they call tangle Floer homology. Ellis, Petkova, and Vértesi [EPV15] show that the dg algebras of [PV14] categorify a tensor product of irreducible representations of U q (gl(1|1)), each of which is V or V * except for one factor L(λ n+1 ) which is neither V nor V * . The Heegaard diagrams motivating these two theories are different; however, it would be interesting to see whether relationships between the theories exist, especially since both theories are known to compute knot Floer homology in some form.…”

Peculiar modules for 4‐ended tangles

“…It is interesting to compare the ideas described in this paper to the combinatorial tangle Floer theory by Petkova and Vértesi and the algebraic tangle homology theory by Ozsváth and Szabó as well as their corresponding decategorifications in terms of the representation theory of . In fact, the definition of our generalised peculiar modules is primarily inspired by the invariants from , see Remark .…”