Unknotting numbers for handlebody-knots and Alexander quandle colorings

Abstract

A crossing change of a handlebody-knot is that of a spatial graph representing it. We see that any handlebody-knot can be deformed into trivial one by some crossing changes. So we define the unknotting numbers for handlebody-knots. In the case classical knots, which are considered as genus one handlebody-knots, Clark, Elhamdadi, Saito and Yeatman gave lower bounds of the Nakanishi indices by the numbers of some finite Alexander quandle colorings, and hence they also gave lower bounds of the unknotting numbers. In this paper, we give lower bounds of the unknotting numbers for handlebody-knots with any genus by the numbers of some finite Alexander quandle colorings of type at most [Formula: see text].