On the transient number of a knot

Mario Eudave-Muñoz and Joan Carlos Segura-Aguilar

Vol. 332 (2024), No. 1, 69–89

DOI: 10.2140/pjm.2024.332.69

Abstract

The transient number of a knot $K$, denoted $\mathrm{tr}<!--FUNCTION\; APPLICATION-->(K)$, is the minimal number of simple arcs that have to be attached to $K$, in order for $K$ to be homotoped to a trivial knot in a regular neighborhood of the union of $K$ and the arcs. We give a lower bound for $\mathrm{tr}<!--FUNCTION\; APPLICATION-->(K)$ in terms of the rank of the first homology group of the double branched cover of $K$. In particular, if $\mathrm{tr}<!--FUNCTION\; APPLICATION-->(K)=1$, then the first homology group of the double branched cover of $K$ is cyclic. Using this, we can calculate the transient number of many knots in the tables and show that there are knots with arbitrarily large transient number.

Keywords

knot, transient number, unknotting number, tunnel number, double branched covers

Mathematical Subject Classification

Primary: 57K10

Secondary: 57M12

Milestones

Received: 26 July 2023

Revised: 4 August 2024

Accepted: 6 September 2024

Published: 20 November 2024