Equivariant rho-invariants and instanton homology of torus knots

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 $q$.

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Received 26 October 2016

Accepted 2 June 2017

Published 12 December 2019