Knot mutation: 4-genus of knots and algebraic concordance

Details Knot mutation: 4-genus of knots and algebraic concordance Knot mutation: 4-genus of knots and algebraic concordance

2003-10-07

arxiv.org/abs/math/0310100v1

Pacific J. Math. 220 (2005), no. 1, 87-105

Mathematics

Geometric Topology

16 pages, 6 figures

Scientific paper

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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