On the transient number of a knot

TL;DR

This paper introduces the transient number of a knot, providing bounds and calculations for various knots, and explores its relationship with the homology of the double branched cover, revealing knots with arbitrarily large transient numbers.

Contribution

It establishes a lower bound for the transient number based on the homology of the double branched cover and computes this number for many knots, showing its unbounded nature.

Findings

Lower bound for transient number in terms of homology

Transient number equals 1 implies cyclic homology

Existence of knots with arbitrarily large transient number

Abstract

The transient number of a knot K, denoted tr(K), is the minimal number of simple arcs that have to be attached to K, in order that K can be homotoped to a trivial knot in a regular neighborhood of the union of K and the arcs. We give a lower bound for tr(K) in terms of the rank of the first homology group of the double branched cover of K. In particular, if t(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.