Skein relations for tangle Floer homology

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 $T$ a differential graded bimodule $\widetilde{\mathrm{CT}}(T)$. If $L$ is obtained by gluing together $T_1,\cdots ,T_m$, then the knot Floer homology $\widehat{\mathrm{HFK}}(L)$ of $L$ can be recovered from $\widetilde{\mathrm{CT}}(T_1),\cdots ,\widetilde{\mathrm{CT}}(T_m)$. In the present paper, we prove combinatorially that tangle Floer homology satisfies unoriented and oriented skein relations, which are analogues of the skein exact triangles for knot Floer homology.