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Equivariant Rho-Invariants and Instanton Homology of Torus Knots
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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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Traceless $\mathrm{SU}(2)$ characters and $\mathbb{Z}/4$ instanton gradings for two-bridge and $(3,n)$-torus knots
For two-bridge knots every traceless SU(2) character is binary-dihedral; for (3,n)-torus knots the characters are mostly non-dihedral, with gradings that predict when knot-instanton homology shrinks below the chain complex.
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