An unknotting index for virtual knots

TL;DR

This paper introduces an unknotting index for virtual knots, providing computational methods, exploring its relationship with virtual knot modules, and demonstrating the existence of knots with specific unknotting indices.

Contribution

It defines the unknotting index for virtual knots and establishes its properties, including existence results and connections to virtual knot invariants.

Findings

Unknotting index can be computed using writhe invariants.

There exist virtual knots with unknotting index (1,n) for any non-negative integer n.

The unknotting index relates to the virtual knot module.

Abstract

In this paper we introduce the notion of an unknotting index for virtual knots. We give some examples of computation by using writhe invariants, and discuss a relationship between the unknotting index and the virtual knot module. In particular, we show that for any non-negative integer $n$ there exists a virtual knot whose unknotting index is $(1,n)$.