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.

arXiv comment5 pages