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Knot mutation: 4-genus of knots and algebraic concordance

Se-Goo Kim·Charles Livingston
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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 construct examples to show 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 of this paper 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. The paper concludes 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.

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Se-Goo Kim, Charles Livingston. 2003-10-07. Knot mutation: 4-genus of knots and algebraic concordance. https://arxiv.org/abs/math/0310100

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