The KW Equations and the Nahm Pole Boundary Condition with Knots

TL;DR

This paper extends the analysis of KW equations with Nahm pole boundary conditions to include knots, proving ellipticity and describing solution behavior near boundaries and knots, advancing the mathematical understanding of knot invariants.

Contribution

It generalizes the Nahm pole boundary condition to knots, proves ellipticity of the KW equations with these conditions, and analyzes solution asymptotics near boundaries and knots.

Findings

KW equations with generalized Nahm pole conditions are elliptic

Solutions are polyhomogeneous near boundary and knots

Indicial equations determine solution exponents

Abstract

It is conjectured that the coefficients of the Jones polynomial can be computed by counting solutions of the KW equations on a four-dimensional half-space, with certain boundary conditions that depend on a knot. The boundary conditions are defined by a "Nahm pole" away from the knot with a further singularity along the knot. In a previous paper, we gave a precise formulation of the Nahm pole boundary condition in the absence of knots, in the present paper, we do this in the more general case with knots included. We show that the KW equations with generalized Nahm pole boundary conditions are elliptic, and that the solutions are polyhomogeneous near the boundary and near the knot, with exponents determined by solutions of appropriate indicial equations. This involves the analysis of a "depth two incomplete iterated edge operator." As in our previous paper, a key ingredient in the analysis is a convenient new Weitzenb\"ock formula that is well-adapted to the specific problem.}