The $\mathbb{Z}/2$-equivariant Seiberg--Witten Floer cohomology of the $(2, 1)$-cable is an unknot detector

Abstract

We prove that a knot $K \subset S^3$ is the unknot if and only if the $\mathbb{Z}/2$-equivariant Seiberg--Witten Floer cohomology of the double cover branched along the $(2, 1)$-cable of $K$ has no $\mathbb{F}_2[Q]$-torsion. The proof combines the Thom--Gysin exact sequence, the Montesinos trick, and the genus bound for $L$-space surgeries.