The pillowcase and traceless representations of knot groups II: a Lagrangian-Floer theory in the pillowcase

TL;DR

This paper develops a Lagrangian-Floer chain complex for immersions into the pillowcase, applying it to knot theory via traceless $SU(2)$ character varieties, and explores its relation to instanton homology.

Contribution

It introduces a new Lagrangian-Floer theory for the pillowcase and connects it to knot invariants and instanton homology, providing computational methods for torus knots.

Findings

Constructed a $Z/4$ graded Lagrangian-Floer chain complex.

Applied the theory to traceless $SU(2)$ character varieties of knots.

Linked the theory to reduced instanton homology of knots.

Abstract

We define an elementary relatively $\mathbb Z/4$ graded Lagrangian-Floer chain complex for restricted immersions of compact 1-manifolds into the pillowcase, and apply it to the intersection diagram obtained by taking traceless $SU(2)$ character varieties of 2-tangle decompositions of knots. Calculations for torus knots are explained in terms of pictures in the punctured plane. The relation to the reduced instanton homology of knots is explored.