Trace-free characters and abelian knot contact homology II

TL;DR

This paper demonstrates that certain torus knots have ghost characters, providing counterexamples to Ng's conjecture which links abelian knot contact homology to character varieties, thus challenging previous assumptions.

Contribution

It introduces ghost characters for specific torus knots and shows how these serve as counterexamples to Ng's conjecture, expanding understanding of knot contact homology.

Findings

Torus knots (4,5) and (5,6) admit ghost characters.

Ghost characters serve as counterexamples to Ng's conjecture.

Ng's conjecture holds for 2-bridge and 3-bridge knots but fails for these torus knots.

Abstract

We show that the $(4,5)$- and $(5,6)$-torus knots admit ghost characters. Consequently, these knots provide counterexamples to Ng's conjecture, which proposes an isomorphism between the complexification of degree $0$ abelian knot contact homology and the coordinate ring of the character variety of the $2$-fold branched cover of the $3$-sphere branched along a knot. While Ng's conjecture has been verified for all $2$-bridge and $3$-bridge knots, we demonstrate, via ghost characters, how this isomorphism fails for these torus knots.