4-Move distance of knots

Abstract

A [Formula: see text]-move is a local change for knots and links which changes 4 half twists to 0 half twists or vice versa. In 1979, Yasutaka Nakanishi conjectured that every knot can be changed by [Formula: see text]-moves to the unknot. This is still an open problem. In this paper, we consider the [Formula: see text]-move distance of knots, which is the minimal number of [Formula: see text]-moves needed to deform one into the other. In particular, the [Formula: see text]-move unknotting number of a knot is the [Formula: see text]-move distance to the unknot. We give a table of the [Formula: see text]-move unknotting number of knots with up to nine crossings.