On the modular Jones polynomial

Guillaume Pagel

doi:10.5802/crmath.106

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~$3^k$.

Highlights

One of the major aims of knot theory is to determine as as possible whether a given knot is isotopic to the unknot
The Jones polynomial is a knot invariant living in the ring of Laurent polynomials over the integers
A long-standing question is to determine whether the Jones polynomial can detect the unknot, meaning that the unknot is the only knot with Jones polynomial equal to 1

Summary

Introduction

One of the major aims of knot theory is to determine as as possible whether a given knot is isotopic to the unknot. Thanks to the modulo operation, some of the coefficients of the polynomial will disappear, and sometimes the Jones polynomial modulo an integer m will become trivial. Nontrivial knots with this property will be called m-trivial. If there exists an m-trivial knot for some m ≥ 2, for all r ≥ 1 there exists infinitely many pairwise distinct mr -trivial knots This result allows us to give a positive answer to Problem 2 for integers of the form 3r , and gives a new proof for m = 2r.

Knots and the Jones polynomial

Proof and consequences of Theorem 3

Arithmetic properties

Further work