Recursion relations and BPS-expansions in the HOMFLY-PT skein of the solid torus

Abstract

Inspired by the skein-valued open Gromov–Witten theory of Ekholm and Shende and the Gopakumar–Vafa formula, we associate to each pair of non-negative integers $(g,l)$ a formal power series with values in the HOMFLY-PT skein of a disjoint union of $l$ solid tori. The formal power series can be thought of as open BPS-states of genus $g$ with $l$ boundary components and reduces to the contribution of a single BPS-state of genus $g$ for $l=0$. Using skein theoretic methods, we show that the formal power series satisfy gluing identities and multi-cover skein relations corresponding to an elliptic boundary node of the underlying curves. For $(g,l)=(0,1)$, we prove a crossing formula which is the multi-cover skein relation corresponding to a hyperbolic boundary node, also known as the pentagon identity.