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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 = 2r with r ≥ 1 and m = 3. Here we show that if the answer is negative for some m, then it is negative for mr with any r ≥ 1. In particular, for any r ≥ 1, we construct nontrivial knots whose Jones polynomial is trivial modulo 3r. [ABSTRACT FROM AUTHOR]

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 $3^r$.