The skein valued mirror of the topological vertex

TL;DR

This paper introduces a skein-valued counting method for holomorphic curves in complex 3-space with boundary conditions, connecting topological string theory and knot invariants through algebraic equations.

Contribution

It develops skein-valued operator equations that uniquely determine the curve counts and relate them to the topological vertex, bridging geometry and knot theory.

Findings

Derived skein-valued operator equations that annihilate the curve count

Proved the equations determine the count uniquely

Confirmed agreement with the topological vertex from string theory

Abstract

We count holomorphic curves in complex 3-space with boundaries on three special Lagrangian solid tori. The count is valued in the HOMFLYPT skein module of the union of the tori. Using 1-parameter families of curves at infinity, we derive three skein valued operator equations which must annihilate the count, and which dequantize to a mirror of the geometry. We show algebraically that the resulting equations determine the count uniquely, and that the result agrees with the topological vertex from topological string theory.