Ribbon-move-unknotting-number-two 2-knots, pass-move-unknotting-number-two 1-knots, and high dimensional analogue

TL;DR

This paper investigates the unknotting numbers related to various local-moves on knots across different dimensions, demonstrating the unboundedness of these invariants and constructing specific examples with prescribed unknotting properties.

Contribution

It introduces the concept of unknotting numbers for local-moves on n-dimensional knots and proves their unboundedness, providing explicit examples across dimensions.

Findings

Existence of 2-knots with ribbon-move-unknotting-number two.

Unboundedness of the ribbon-move-unknotting-number for 2-knots.

Construction of 1-knots with pass-move-unknotting-number greater than any given n.

Abstract

The (ordinary) unknotting-number of 1-dimensional knots, which is defined by using the crossing-change, is a very basic and important invariant. It is very natural to consider the `unknotting-number' associated with other local-moves on n-dimensional knots, where n is a natural number. In this paper we prove the following facts. For the ribbon-move on 2-knots, which is a kind of local-move on knots, we have the following: There is a ribbon-move-unknotting-number-two 2-knot. The ribbon-move-unknotting-number of 2-knots is unbounded. For the pass-move on 1-knots, which is a kind of local-move on knots, we have the following: There is a pass-move-unknotting-number-two 1-knot whose (ordinary) unknotting-number is 4. For any natural number n, there is a 1-knot whose pass-move-unknotting-number is>n and whose (ordinary) unknotting-number is 4n. For the high-dimensional-pass-move on high-dimensional knots, which is a kind of local-move on knots, we have the following: There is a (2k+1, 2k+2)-pass-move-unknotting-number-two (4k+2)-knot. The (2k+1, 2k+2)-pass-move-unknotting-number of (4k+2)-knot is unbounded. There is a (2k+1, 2k+1)-pass-move-unknotting-number-two (4k+1)-knot. The (2k+1, 2k+1)-pass-move-unknotting-number of (4k+1)-knot is unbounded. There is a (4k+1)-knot whose twist-move-unknotting-number is n for any natural number n.