Equivariant rho-invariants and instanton homology of torus knots

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.where θ : π 1 (Y ) → U(n) is the trivial representation. It is independent of the metric and defines a diffeomorphism invariant of (Y, α).This paper deals with equivariant analogues of the η-and ρ-invariants in presence of an orientation preserving isometric involution τ : Y → Y as defined in [APS75] and [Don78]. Suppose that a representation α : π 1 (Y ) → U(n) is such that its pull-back τ * α is conjugate to α. Then the pull-back of the flat vector bundle E α is isomorphic to E α hence τ can be lifted to a flat bundle automorphism ν : E α → E α . Note that the lift ν need not be an involution and that there may be more than one lift. The lift ν acts on Ω ev (Y ; E α ) via pull-back of forms making the diagramcommute. Each eigenspace W λ,α of the operator B ev α is then ν * -invariant, and the function

Chern-Simons functional, singular instantons, and the four-dimensional clasp number

“…Another strategy to prove Corollary 3 involves directly computing the mod 4 gradings of the singular flat connections. Anvari computed these mod 4 gradings in terms of certain arithmetic functions and verified Corollary 3 for the p3, 6k `1q torus knots [Anv19]. Corollary 3 has implications about the arithmetic functions appearing in [Anv19].…”

“…Anvari computed these mod 4 gradings in terms of certain arithmetic functions and verified Corollary 3 for the p3, 6k `1q torus knots [Anv19]. Corollary 3 has implications about the arithmetic functions appearing in [Anv19].…”