The pillowcase and perturbations of traceless representations of knot groups

TL;DR

This paper develops explicit holonomy perturbations for flat connection moduli spaces on 3-balls with arcs, enabling detailed study of singular instanton knot homology and computations for specific knots via pillowcase representations.

Contribution

It introduces a concrete local perturbation method for moduli spaces of flat singular SO(3) connections, facilitating the analysis of instanton homology and its relation to knot invariants.

Findings

Computed singular instanton homology for various torus knots.

Analyzed the image of traceless representations in the pillowcase.

Gained insights into differentials in the spectral sequence from Khovanov to instanton homology.

Abstract

We introduce explicit holonomy perturbations of the Chern-Simons functional on a 3-ball containing a pair of unknotted arcs. These perturbations give us a concrete local method for making the moduli spaces of flat singular SO(3) connections relevant to Kronheimer and Mrowka's singular instanton knot homology non-degenerate. The mechanism for this study is a (Lagrangian) intersection diagram which arises, through restriction of representations, from a tangle decomposition of a knot. When one of the tangles is trivial, our perturbations allow us to study isolated intersections of two Lagrangians to produce minimal generating sets for singular instanton knot homology. The (symplectic) manifold where this intersection occurs corresponds to the traceless character variety of the four-punctured 2-sphere, which we identify with the familiar pillowcase. We investigate the image in this pillowcase of the traceless representations of tangles obtained by removing a trivial tangle from 2-bridge knots and torus knots. Using this, we compute the singular instanton homology of a variety of torus knots. In many cases, our computations allow us to understand non-trivial differentials in the spectral sequence from Khovanov homology to singular instanton homology.