VIRTUAL KNOTS AND LINKS WITH UNKNOTTING INDEX (n,m)

In [9], K. Kaur 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).