On the transient number of a knot
Geometric Topology
2024-11-27 v1
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.
Cite
@article{arxiv.2307.14622,
title = {On the transient number of a knot},
author = {Mario Eudave-Muñoz and Joan Carlos Segura Aguilar},
journal= {arXiv preprint arXiv:2307.14622},
year = {2024}
}
Comments
21 pages, 3 figures