The worldsheet skein D-module and basic curves on Lagrangian fillings of the Hopf link conormal

TL;DR

This paper develops a new skein-theoretic D-module framework for understanding HOMFLYPT polynomials via holomorphic curves on knot conormals, with explicit results for the Hopf link conormal.

Contribution

It introduces a worldsheet skein module and D-module approach that unifies skein curve counts and provides explicit generators for the Hopf link conormal case.

Findings

The skein D-module for the Hopf link conormal is generated by three operator polynomials.

Partition functions for Lagrangian fillings satisfy recursive relations and admit quiver-like expansions.

The framework applies to any Lagrangian filling, capturing all holomorphic curve contributions.

Abstract

HOMFLYPT polynomials of knots in the 3-sphere in symmetric representations satisfy recursion relations. Their geometric origin is holomorphic curves at infinity on knot conormals that determine a $D$-module with characteristic variety the Legendrian knot conormal augmention variety and with the recursion relations as operator polynomial generators [arXiv:1304.5778, arXiv:1803.04011]. We consider skein lifts of recursions and $D$-modules corresponding to skein valued open curve counts [arXiv:1901.08027] that encode HOMFLYPT polynomials colored by arbitrary partitions. We define a worldsheet skein module which is the universal target for skein curve counts and a corresponding $D$-module. We then consider the concrete example of the Legendrian conormal of the Hopf link. We show that the worldsheet skein $D$-module for the Hopf link conormal is generated by three operator polynomials that annihilate the skein valued partition function for any choice of Lagrangian filling and recursively determine it uniquely. We find Lagrangian fillings for any point in the augmentation variety and show that their skein valued partition functions admit quiver-like expansions where all holomorphic curves are generated by a small number of basic holomorphic disks and annuli and their multiple covers.