Cosmetic crossing changes of fibered knots (Q2910999)

For a knot, a crossing change is a basic operation. This operation is described as some Dehn surgery along the boundary circle \(\partial D\) of an embedded disk \(D\) in the knot complement which meets the knot transversely in two points with opposite signs. In most cases, a crossing change will change its knot type. On the other hand, if \(\partial D\) bounds another embedded disk in the knot complement, then the crossing change along \(\partial D\) does not change the knot type. In this case, the corresponding crossing is said to be nugatory. Problem 1.58 of [Kazez, William H. (ed.), Geometric topology. 1993 Georgia international topology conference, August 2--13, 1993, Athens, GA, USA. Providence, RI: American Mathematical Society. AMS/IP Stud. Adv. Math. 2(pt.2), 35--473 (1997; Zbl 0888.57014)], asks if a crossing change on a knot yields a knot isotopic to the original one is the crossing nugatory. For the unknot, this is true by \textit{D. Gabai} [J. Differ. Geom. 18, 445--503 (1983; Zbl 0533.57013)]. Also, \textit{I. Torisu} [Topology Appl. 92, No. 2, 119--129 (1999; Zbl 0926.57004)] solved it for two-bridge knots. The main result of the paper under review is to solve the above problem affirmatively for fibered knots. In fact, the author proves this for a generalized crossing change. The proof uses essentially a result of \textit{D. Kotschick} [Proc. Am. Math. Soc. 132, No. 11, 3167--3175 (2004; Zbl 1055.20037)] on a commutator length in the mapping class group of a surface.NEWLINENEWLINEThe final part discusses the adjacency introduced by \textit{E. Kalfagianni} and \textit{X.-S. Lin} [Pac. J. Math. 228, No. 2, 251--275 (2006; Zbl 1123.57007)]. It is proved that if a knot \(K\) is \(n\)-adjacent to a fibered knot \(K'\) for some \(n>1\), then the genus of \(K\) is larger than that of \(K'\) or \(K\) is isotopic to \(K'\).