Abstract

Kearton observed that mutation can change the concordance class of a knot. A close examination of his example reveals that it is of 4-genus 1 and has a mutant of 4-genus 0. The first goal of this paper is to show by examples that for any pair of nonnegative integers m and n there is a knot of 4-genus m with a mutant of 4-genus n.

A second result is a crossing change formula for the algebraic concordance class of a knot, which is then applied to prove the invariance of the algebraic concordance class under mutation. We conclude with an application of crossing change formulas to give a short new proof of Long’s theorem that strongly positive amphicheiral knots are algebraically slice.

Keywords

mutation, knot concordance, amphicheiral, 4-genus, knot genus

Mathematical Subject Classification

Primary: 57M25

Authors