Strongly invertible knots, equivariant slice genera, and an equivariant algebraic concordance group

TL;DR

This paper develops new lower bounds on the equivariant slice genus of strongly invertible knots using the Blanchfield form, introduces an equivariant algebraic concordance group, and reveals the infinite rank of its kernel.

Contribution

It introduces an equivariant algebraic concordance group and establishes lower bounds on equivariant slice genus for strongly invertible knots, expanding understanding of knot concordance.

Findings

Lower bound of n/4 on equivariant slice genus for certain knots

Infinite rank of the kernel of the forgetful map in the equivariant algebraic concordance group

Application of the Blanchfield form to equivariant knot invariants

Abstract

We use the Blanchfield form to obtain a lower bound on the equivariant slice genus of a strongly invertible knot. For our main application, let $K$ be a genus one strongly invertible slice knot with nontrivial Alexander polynomial. We show that the equivariant slice genus of an equivariant connected sum $\#^n K$ is at least $n/4$. We also formulate an equivariant algebraic concordance group, and show that the kernel of the forgetful map to the classical algebraic concordance group is infinite rank.