Band surgery on knots and links, II (Q2890232)

An oriented 2-component link \(L\) is called band-trivializable, if \(L\) can be unknotted by a single band surgery. In this article the author completely determines the band-trivializability for prime links up to 9 crossings. This paper is a continuation of the author's study of band surgery [J. Knot Theory Ramifications 19, No. 12, 1535--1547; (2010; Zbl 1213.57011)] where he discussed when two links are related by band surgery. (Note that the number of components of a link changes when performing band surgery so we use the word links to mean knot or link.)NEWLINENEWLINELet \(L\) and \(L'\) be two links related by a single band surgery. The author shows that the band surgery relationship implies specific relationships between the values of various knot invariants of \(L\) and \(L'\). If these relationships do not hold between the values of one of these knot invariants then \(L\) and \(L'\) cannot be related by a single band surgery. The knot invariants used here are the 4-ball genus, the Arf invariant, the signature and various different knot polynomials. Finally, the author addresses the questions whether a \((2n+1)\)-crossing two bridge knot is related to a \((2,2n)\)-torus link by a band surgery for the values of \(n=3, 4\) which is a question arising from the study of site-specific recombination experiments involving knotted DNA molecules.