Trace-free characters and abelian knot contact homology II

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.