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arXiv · 2307.14622

On the transient number of a knot

Mario Eudave-Muñoz·Joan Carlos Segura Aguilar
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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.

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Mario Eudave-Muñoz, Joan Carlos Segura Aguilar. 2023-07-27. On the transient number of a knot. https://doi.org/10.2140/pjm.2024.332.69

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