An infinite family of prime knots with a certain property for the clasp number

TL;DR

This paper constructs an infinite family of prime knots demonstrating that the clasp number can exceed both the genus and unknotting number, answering a fundamental question about the relationship between these knot invariants.

Contribution

It proves the existence of an infinite family of prime knots where the clasp number is strictly greater than both the genus and unknotting number.

Findings

Existence of prime knots with clasp number greater than genus and unknotting number

Construction method for such infinite family of knots

Clarification of the relationship between clasp number, genus, and unknotting number

Abstract

The clasp number $c(K)$ of a knot $K$ is the minimum number of clasp singularities among all clasp disks bounded by $K$. It is known that the genus $g(K)$ and the unknotting number $u(K)$ are lower bounds of the clasp number, that is, $\max\{g(K),u(K)\} \leq c(K)$. Then it is natural to ask whether there exists a knot $K$ such that $\max\{g(K),u(K)\}<c(K)$. In this paper, we prove that there exists an infinite family of prime knots such that the question above is affirmative.