Knot Floer homology and surgery on equivariant knots

Abstract

Given an equivariant knot K of order 2, we study the induced action of the symmetry on the knot Floer homology. We relate this action with the induced action of the symmetry on the Heegaard Floer homology of large surgeries on K. This surgery formula can be thought of as an equivariant analog of the involutive large surgery formula proved by Hendricks and Manolescu. As a consequence, we obtain that for certain double branched covers of S 3 and corks, the induced action of the involution on Heegaard Floer homology can be identified with an action on the knot Floer homology. As an application, we calculate equivariant correction terms which are invariants of a generalized version of the spin rational homology cobordism group, and define two knot concordance invariants. We also compute the action of the symmetry on the knot Floer complex of K for several equivariant knots.

The $(2,1)$-cable of the figure-eight knot is not smoothly slice

“…This approach was employed successfully in [AKS20, DHM22] using tools from Heegaard Floer homology [OS04c,OS04b]. Building on recent results from [DMS22,Mal22], we extend these methods to prove: Theorem 1.1. Let K = 4 1 be the figure-eight knot.…”