Recursion relations and BPS-expansions in the HOMFLY-PT skein of the solid torus
Lukas Nakamura
TL;DR
This work builds a skein-valued open Gromov-Witten framework for the HOMFLY-PT skein of solid tori, introducing BPS partition functions $\Psi^{\pm(g,l)}$ that encode genus-$g$ and boundary-component-$l$ data. It proves foundational recursion and gluing identities, including a crossing formula linking $\Psi^{(g,l)}$ to $\Psi^{(g,l-1)}$ and exponentiated disk/annulus cases that recover known Gromov-Witten building blocks, thereby providing a skein-theoretic realization of multi-cover phenomena. The paper derives explicit closed forms for low-genus cases (e.g., $\Psi^{(1,1)}$, $\Psi^{(0,1)}$, $\Psi^{(0,2)}$) and proves general gluing formulas expressed via dual bases $W_\lambda^*$, enabling composite constructions from basic disks and annuli. Collectively, these results tie skein-theoretic recursion to Gopakumar–Vafa-type decompositions and LMOV-type integrality data, offering a calculational framework with potential applications to open string invariants and topological quantum field theory.
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.