On the modular Jones polynomial
A major problem in knot theory is to decide whether the Jones polynomial detects the unknot. In this paper we study a weaker related problem, namely whether the Jones polynomial reduced modulo an integer $m$ detects the unknot. The answer is known to be negative for $m=2^r$ with $r\ge 1$ and $m=3$....
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| Format: | Article |
| Language: | English |
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Académie des sciences
2020-12-01
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| Series: | Comptes Rendus. Mathématique |
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| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.106/ |
| _version_ | 1797651546127728640 |
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| author | Pagel, Guillaume |
| author_facet | Pagel, Guillaume |
| author_sort | Pagel, Guillaume |
| collection | DOAJ |
| description | A major problem in knot theory is to decide whether the Jones polynomial detects the unknot. In this paper we study a weaker related problem, namely whether the Jones polynomial reduced modulo an integer $m$ detects the unknot. The answer is known to be negative for $m=2^r$ with $r\ge 1$ and $m=3$. Here we show that if the answer is negative for some $m$, then it is negative for $m^r$ with any $r\ge 1$. In particular, for any $r\ge 1$, we construct nontrivial knots whose Jones polynomial is trivial modulo $3^r$. |
| first_indexed | 2024-03-11T16:17:21Z |
| format | Article |
| id | doaj.art-7cc0d03726f04e3496b62326d41dc156 |
| institution | Directory Open Access Journal |
| issn | 1778-3569 |
| language | English |
| last_indexed | 2024-03-11T16:17:21Z |
| publishDate | 2020-12-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj.art-7cc0d03726f04e3496b62326d41dc1562023-10-24T14:18:50ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692020-12-01358890190810.5802/crmath.10610.5802/crmath.106On the modular Jones polynomialPagel, Guillaume0Univ. Littoral Côte d’Opale, UR 2597, LMPA, Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville, F-62100 Calais, FranceA major problem in knot theory is to decide whether the Jones polynomial detects the unknot. In this paper we study a weaker related problem, namely whether the Jones polynomial reduced modulo an integer $m$ detects the unknot. The answer is known to be negative for $m=2^r$ with $r\ge 1$ and $m=3$. Here we show that if the answer is negative for some $m$, then it is negative for $m^r$ with any $r\ge 1$. In particular, for any $r\ge 1$, we construct nontrivial knots whose Jones polynomial is trivial modulo $3^r$.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.106/KnotJones polynomialKauffman bracket<span class="mathjax-formula">$m$</span>-trivial knotconnected sumLegendre formulamodular arithmetic |
| spellingShingle | Pagel, Guillaume On the modular Jones polynomial Comptes Rendus. Mathématique Knot Jones polynomial Kauffman bracket <span class="mathjax-formula">$m$</span>-trivial knot connected sum Legendre formula modular arithmetic |
| title | On the modular Jones polynomial |
| title_full | On the modular Jones polynomial |
| title_fullStr | On the modular Jones polynomial |
| title_full_unstemmed | On the modular Jones polynomial |
| title_short | On the modular Jones polynomial |
| title_sort | on the modular jones polynomial |
| topic | Knot Jones polynomial Kauffman bracket <span class="mathjax-formula">$m$</span>-trivial knot connected sum Legendre formula modular arithmetic |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.106/ |
| work_keys_str_mv | AT pagelguillaume onthemodularjonespolynomial |