# On the modular Jones polynomial

Abstract : 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 $n$ detects the unknot. The answer is known to be negative for $n=2^k$ with $k\geq 1$ and $n=3$. Here we show that if the answer is negative for some $n$, then it is negative for $n^k$ with any $k\geq 1$. In particular, for any $k\geq 1$, we construct nontrivial knots whose Jones polynomial is trivial modulo~$3^k$.
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https://hal.archives-ouvertes.fr/hal-02909973
Contributor : Guillaume Pagel Connect in order to contact the contributor
Submitted on : Thursday, December 24, 2020 - 2:15:20 PM
Last modification on : Monday, May 2, 2022 - 1:10:02 PM

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Guillaume Pagel. On the modular Jones polynomial. Comptes Rendus. Mathématique, Académie des sciences (Paris), 2020, 358 (8), pp.901-908. ⟨10.5802/crmath.106⟩. ⟨hal-02909973v2⟩

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