Knot mutation: 4-genus of knots and algebraic concordance
The 4-genus of a knot \(K\), \(g_4(K)\), is the least genus of an embedded surface bounded by \(K\) in the 4-ball. One of the results of this paper is the following: Theorem. For every pair of nonnegative integers \(m\) and \(n\), there is a knot \(K\) with mutant \(K^*\) satisfying \(g_4(K)=m\) and \(g_4(K^*)=n\). This is shown by examples which are constructed using \textit{C. Kearton}'s observation [Proc. Am. Math. Soc. 105, 206--208 (1989; Zbl 0675.57005)]. Next, the authors develop a crossing change formula for the algebraic concordance class of a knot, and show that it is presented by mutation. Finally, the authors apply the crossing change formula to give a new proof of \textit{D. D. Long}'s theorem [Math. Proc. Camb. Philos. Soc. 95, 309--312 (1984; Zbl 0547.57006)] that strongly positive amphicheiral knots are algebraically slice.
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