Skein traces and curve counting

Skein modules are vector space-valued invariants of 3-manifolds describing the space of line defects modulo skein relations (determined by a choice of a ribbon category). When the 3-manifold is S x I for some surface S, the skein module has a natural algebra structure and is called the skein algebra of S.
In 2010, Bonahon and Wong constructed an algebra embedding (named “quantum trace”) of the sl_2 skein algebra into a quantum cluster variety called the “quantum Teichmuller space” for punctured surfaces, which has applications to the representation theory of skein algebras.
In the first half of this talk, I will give an overview of these concepts and explain how the quantum trace map can be generalized to the 3-dimensional setup.
In the second half, I will discuss how everything above can be generalized to HOMFLYPT skeins and has natural interpretation in terms of counts of holomorphic curves.