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Quantum Topology

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Volume 11, Issue 1, 2020, pp. 119–225
DOI: 10.4171/QT/134

Published online: 2020-02-24

Skein relations for tangle Floer homology

Ina Petkova[1] and C.-M. Michael Wong[2]

(1) Dartmouth College, Hanover, USA
(2) Louisiana State University, Baton Rouge, USA

In a previous paper, VĂ©rtesi and the first author used grid-like Heegaard diagrams to define tangle Floer homology, which associates to a tangle $T$ a differential graded bimodule $\tilde{\mathrm {CT}} (T)$. If $L$ is obtained by gluing together $T_1, \cdots, T_m$, then the knot Floer homology $\hat{\mathrm {HFK}} (L)$ of $L$ can be recovered from $\tilde{\mathrm {CT}}(T_1), \cdots, \optilde{\CT} (T_m)$. 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.

Keywords: Tangles, knot Floer homology, bordered Floer homology, skein relations

Petkova Ina, Wong C.-M. Michael: Skein relations for tangle Floer homology. Quantum Topol. 11 (2020), 119-225. doi: 10.4171/QT/134