An unoriented skein relation via bordered-sutured Floer homology (Q2673139)

Knot Floer homology is an invariant of oriented, null-homologous knots \(K\) in an oriented, connected, closed \(3\)-manifold \(Y\) which categorifies the Alexander polynomial. Since the Alexander polynomial satisfies an oriented skein relation, the knot Floer homology satisfies an oriented skein exact triangle [\textit{P. Ozsváth} and \textit{Z. Szabó}, Adv. Math. 186, No. 1, 58--116 (2004; Zbl 1062.57019)]. Furthermore it has been proven by \textit{C. Manolescu} [Math. Res. Lett. 14, No. 5--6, 839--852 (2007; Zbl 1161.57005)] that knot Floer homology of a link in \(S^3\) satisfies an unoriented skein relation. Bordered-sutured Floer homology is an invariant of bordered-sutured \(3\)-manifolds which generalizes both bordered Floer homology and sutured Floer homology. It is known that knot Floer homology of a link can be recovered via sutured Floer homology of the link complement. In the paper under review, the authors consider the pair \((Y, T_\infty)\) of a tangle \(T_\infty\) and a compact, oriented \(3\)-manifold (possibly) with boundary \(Y\) which satisfies certain conditions. Fixing a tangle diagram of \(T_\infty\), and a crossing, the authors also consider \((Y, T_0)\) and \((Y,T_1)\), where \(T_0\) and \(T_1\) are the two possible (unoriented) resolutions of the fixed crossing in \(T_\infty\). The main result is that there are homomorphisms giving an exact triangle relating the bordered-sutured Floer homology of \((Y,T_0)\), \((Y,T_1)\) and \((Y,T_\infty)\). The homomorphisms in the exact triangle are defined by counts of bordered holomorphic polygons, and these counts are also described combinatorially by the authors. In the special case of \(Y = S^3\) the exact triangle specializes to Manolescu's exact triangle. Finally the authors remark that the same strategy applies to a more general setting. It can e.g.~be used to prove an unoriented skein exact triangle for (a generalization of) graph Floer homology.