On the transient number of a knot
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866929604344676352 |
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| author | Eudave-Muñoz, Mario Aguilar, Joan Carlos Segura |
| author_facet | Eudave-Muñoz, Mario Aguilar, Joan Carlos Segura |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2307_14622 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the transient number of a knot Eudave-Muñoz, Mario Aguilar, Joan Carlos Segura Geometric Topology 57K10, 57M12 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. |
| title | On the transient number of a knot |
| topic | Geometric Topology 57K10, 57M12 |
| url | https://arxiv.org/abs/2307.14622 |