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 $m$ 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 $3^r$.

Keywords

• Knot
• Jones polynomial
• Kauffman bracket
• $m$-trivial knot%22%2C%22default_operator%22%3A%22AND%22%7D%7D%7D"> <span class="mathjax-formula">$m$</span>-trivial knot
• connected sum
• Legendre formula
• modular arithmetic