Skein relations for tangle Floer homology

TL;DR

This paper establishes skein relations for tangle Floer homology, enabling combinatorial computation and generalization of known skein exact triangles from knot Floer homology.

Contribution

It proves combinatorial skein relations for tangle Floer homology, extending the framework of skein exact triangles to tangles.

Findings

Proves unoriented skein relations for tangle Floer homology.

Establishes oriented skein relations for tangle Floer homology.

Generalizes skein exact triangles from knot to tangle Floer homology.

Abstract

In a previous paper, V\'ertesi and the first author used grid-like Heegaard diagrams to define tangle Floer homology, which associates to a tangle $T$ a differential graded bimodule $\widetilde{\mathrm{CT}} (T)$. If $L$ is obtained by gluing together $T_1, \dotsc, T_m$, then the knot Floer homology $\hat{\mathrm{HFK}}(L)$ of $L$ can be recovered from $\widetilde{\mathrm{CT}} (T_1), \dotsc, \widetilde{\mathrm{CT}} (T_m)$. 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.