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 $n$ detects the unknot. The answer is known to be negative for $n=2^k$ with $kgeq 1$ and $n=3$. Here we show that if the answer is negative for some $n$, then it is negative for $n^k$ with any $kgeq 1$. In particular, for any $kgeq 1$, we construct nontrivial knots whose Jones polynomial is trivial modulo~$3^k$.