Cosmetic crossings and Seifert matrices (Q357181)

For an oriented knot \(K\) in the \(3\)-sphere, a crossing change is realized by taking a disk \(D\) whose interior meets \(K\) in two points with opposite intersection numbers and twisting \(D\). If the crossing circle \(\partial D\) bounds another disk which is disjoint from the knot, then the crossing change is said to be nugatory. Clearly, a nugatory crossing change does not change the knot type. A non-nugatory crossing change is said to be cosmetic if the resulting knot is isotopic to the original one. The nugatory crossing conjecture asserts that if a crossing change on a knot yields the same knot then the crossing change is nugatory. This conjecture is known to be true for the trivial knot, two-bridge knots, fibered knots.NEWLINENEWLINEThe main purpose of the paper under review is to give necessary conditions for oriented genus one knots to admit cosmetic crossing changes. Let \(K\) be an oriented genus one knot. The authors claim that if \(K\) admits a cosmetic crossing change, then it is algebraically slice, and the first homology group of the double branched cover is finite cyclic. As an immediate corollary, if the determinant is not a perfect square, then a genus one knot admits no cosmetic crossing change. Further, by using Seifert matrices, if \(K\) enjoys the additional property that it has a unique minimal Seifert surface, then the Alexander polynomial is shown to be trivial. Also, the nugatory crossing conjecture is solved for genus one knots with at most 12 crossings, some twisted Whitehead doubles, and some \(3\)-strand pretzel knots.