On the modular Jones polynomial

TL;DR

This paper investigates whether the Jones polynomial, reduced modulo an integer, can detect the unknot, revealing that if it fails for some n, it also fails for all powers of n, and constructs knots with trivial Jones polynomial mod 3^k.

Contribution

It establishes a recursive property of the Jones polynomial modulo integers and constructs explicit examples of knots with trivial polynomial modulo powers of 3.

Findings

If the Jones polynomial mod n does not detect the unknot, then it also fails for all powers of n.

Constructs nontrivial knots with Jones polynomial trivial mod 3^k for any k ≥ 1.

Shows the limitations of Jones polynomial as a knot detector under modular reduction.

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~$3^k$.