  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.
