On amphichirality of symmetric unions

TL;DR

This paper investigates the properties of symmetric unions in knots, proving that certain amphichiral symmetric unions are trivial or decomposable, thus contributing to the understanding of the Jones polynomial and knot triviality.

Contribution

It proves that amphichiral symmetric unions with one twist region are trivial and generalizes the result to unions of any knots, showing they decompose into a knot and its mirror image.

Findings

Amphichiral symmetric unions with one twist are always trivial.

Symmetric unions of any knots with one twist are connected sums of the knot and its mirror.

The results restrict possible counterexamples to the unknotting problem via Jones polynomial.

Abstract

It is still unknown whether there is a nontrivial knot with Jones polynomial equal to that of the unknot. Tanaka shows that if an amphichiral knot is a symmetric union of the unknot with one twist region, then its Jones polynomial is trivial. Hence, he proposes that if any of these knots were nontrivial, a nontrivial knot with trivial Jones polynomial would exist. We first show such a knot is always trivial and hence cannot be used to answer the above question. We then generalize the argument to symmetric unions of any knots and show that if a symmetric union of a knot $J$ with one twist region is amphichiral, then it is the connected sum of $J$ and its mirror image $-J$.