On the Alexander polynomials of knots with Gordian distance one (Q409545)

For a pair of knots, the Gordian distance between them is defined to be the least number of crossing changes needed to obtain one from the other. This is a generalization of the classical unknotting number. The main result of the paper under review is to give a condition for the Alexander polynomials which are realizable by a pair of knots with Gordian distance one. Roughly speaking, the condition is described as the existence of some submodule of the quotient ring \(\Lambda/(a,b)\), where \(\Lambda=\mathbb{Z}[t,t^{-1}]\) is the one-variable integral Laurent polynomial ring and \((a,b)\) is the ideal generated by Alexander polynomials \(a\), \(b\in \Lambda\).NEWLINENEWLINEAs an application, the author gives an infinite set of Alexander polynomials in which any pair is not realizable by knots with Gordian distance one, but realizable by knots with Gordian distance two. In fact, infinitely many, mutually disjoint such sets are given. One of these sets contains both of the Alexander polynomials of the trefoil and the figure-eight knot. This solves a problem by [\textit{Y. Nakanshi}, J. Knot Theory Ramifications 14, No. 1, 3--8 (2005; Zbl 1067.57005)]. Also, it is shown that if one of a pair of Alexander polynomials is realizable by a slice knot, then the pair is realizable by knots with Gordian distance one.