Open strings on knot complements

TL;DR

This paper develops a skein valued holomorphic curve counting approach to compute the partition function of Lagrangian knot complements, connecting it to augmentation curves, representation varieties, and $q$-difference equations for knot invariants.

Contribution

It introduces a new flow loop formula for skein valued partition functions of knot complements and links it to augmentation curves and quantum $q$-difference equations.

Findings

Partition functions localize on holomorphic annuli for torus knots.

The partition function satisfies a $q$-difference equation related to HOMFLYPT polynomials.

Provides a geometric coordinate chart for the quantized augmentation polynomial.

Abstract

Using skein valued holomorphic curve counting techniques, we give a flow loop formula for the skein valued partition function of the Lagrangian knot complement of a fibered knot (of the $A$-model open topological strings with Lagrangian $A$-branes wrapping the complement) in the cotangent bundle of the three-sphere and in the resolved conifold. For torus knots we show that the partition function in the cotangent bundle localizes on two or three holomorphic annuli and give a corresponding generalized quiver structure for the partition function in the resolved conifold. We connect the formula to the augmentation curve, the representation variety of the knot contact homology algebra of the knot, generated by Reeb chords of its Legendrian conormal and with differential given by holomorphic disks interpolating between words of Reeb chords. The curve admits a quantization as a $q$-difference equation for the generating function of symmetrically colored HOMFLYPT-polynomials of the knot or, geometrically, for the $U(1)$-partition function of the knot conormal. For $(2,2p+1)$-torus knots we show that, after a change of variables, the partition function of the knot complement also satisfies this $q$-difference equation. This gives another geometrically defined coordinate chart for the $D$-module defined by the quantized augmentation polynomial.