Equivariant Rho-Invariants and Instanton Homology of Torus Knots

TL;DR

This paper introduces equivariant rho-invariants for 3-manifolds with involutions, computes them for lens spaces and Brieskorn spheres, and applies these results to analyze the instanton homology of torus knots.

Contribution

It extends classical rho-invariants to an equivariant setting, providing explicit computations and applying them to knot Floer homology of torus knots.

Findings

Computed equivariant rho-invariants for lens spaces and Brieskorn spheres.

Determined generators and Floer gradings for instanton chain complexes of (p,q)-torus knots.

Established a link between involutions and instanton homology computations.

Abstract

The equivariant rho-invariants studied in this paper are a version of the classical rho-invariants of Atiyah, Patodi, and Singer in the presence of an isometric involution. We compute these rho-invariants for all involutions on the 3-dimensional lens spaces with 1-dimensional fixed point sets, as well as for some involutions on Brieskorn homology spheres. As an application, we compute the generators and Floer gradings in the singular instanton chain complex of (p,q)-torus knots with odd p and q.