Virtual knots and links with unknotting index $(n,m)$
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In \cite{kaur2019unknottingknot}, K. Kaur {\it 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 $(n,m)$.
Abstract
In \cite{kaur2019unknottingknot}, K. Kaur {\it 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 $(n,m)$.