Virtual knots and links with unknotting index (n,m)

TL;DR

This paper constructs infinite families of virtual knots and links with specified unknotting indices, advancing understanding of their classification and existence for various pairs of non-negative integers.

Contribution

It provides the first explicit constructions of virtual knots and links with arbitrary unknotting indices (n,m), confirming their existence for all such pairs.

Findings

Infinite families of virtual knots with unknotting index (0,m)

Infinite families of virtual knots with unknotting index (1,0)

Existence of virtual knots with any positive integer pair (n,m)

Abstract

In [8], K. Kaur, S. Kamada et al. posed a problem of finding a virtual knot, if exists, with an unknotting index (n,m), where (n,m) is a pair of non-negative integers. In this paper, we address this question by providing infinite families of virtual knots with unknotting indices (0,m) and (1,0), respectively. In general, we establish the existence of infinitely many distinct virtual knot diagrams with unknotting index (n,m), for any pair (n,m) of positive integers. Furthermore, we positively address this question for k(>1)-component virtual links positively by providing infinite families of k(>1)-component virtual links with unknotting index (n,m), for a given pair of non-negative integers (n,m).