The “unknotting number” associated with other local moves than the crossing-change

Abstract

The ordinary unknotting number of 1-dimensional knots, which is defined by using the crossing-change, is a very basic and important invariant. Let [Formula: see text] be a positive integer. It is very natural to consider the “unknotting number” associated with other local moves on [Formula: see text]-dimensional knots. In this paper, we prove the following. For the ribbon-move on 2-knots, which is a local move on knots, we have the following: There is a 2-knot which is changed into the unknot by two times of the ribbon-move not by one time. The “unknotting number” associated with the ribbon-move is unbounded. For the pass-move on 1-knots, which is a local move on knots, we have the following: There is a 1-knot such that it is changed into the unknot by two times of the pass-move not by one time and such that the ordinary unknotting number is [Formula: see text]. For any positive integer [Formula: see text], there is a 1-knot whose “unknotting number” associated with the pass-move is [Formula: see text] and whose ordinary unknotting number is [Formula: see text]. Let [Formula: see text] and [Formula: see text] be positive integers. For the [Formula: see text]-move on [Formula: see text]-knots, which is a local move on knots, we have the following: Let [Formula: see text] be a non-negative integer. There is a [Formula: see text]-knot which is changed into the unknot by two times of the [Formula: see text]-move not by one time. The “unknotting number” associated with the [Formula: see text]-move is unbounded. There is a [Formula: see text]-knot which is changed into the unknot by two times of the [Formula: see text]-move not by one. The “unknotting number” associated with the [Formula: see text]-move is unbounded. We prove the following: For any positive integer [Formula: see text] and any positive integer [Formula: see text], there is a [Formula: see text]-knot which is changed into the unknot by [Formula: see text] times of the twist-move not by [Formula: see text] times.