Knot mutation: 4-genus of knots and algebraic concordance

Se-Goo Kim and Charles Livingston

Vol. 220 (2005), No. 1, 87–106

DOI: 10.2140/pjm.2005.220.87

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 2000

Primary: 57M25

Milestones

Received: 26 November 2003

Revised: 11 May 2004

Published: 1 May 2005