Trace-free characters and abelian knot contact homology I

Abstract

We study the structure underlying Ng's conjecture, which relates the degree $0$ abelian knot contact homology of a knot $K$ to the coordinate ring of the $SL_2(\mathbf{C})$-character variety $X(\Sigma_2 K)$ of the $2$-fold branched cover of the $3$-sphere branched along $K$. Our approach is based on the study of (meridionally) trace-free characters of knot groups. For each knot $K$, they form a closed algebraic subset $S_0(K)$ of the $SL_2(\mathbf{C})$-character variety of $K$, defined by the trace-free condition on meridians. The subset $S_0(K)$, called the trace-free slice of $K$, has a natural connection to $X(\Sigma_2K)$. We show that the trace-free slice admits the structure of a $2$-fold branched cover of a closed algebraic set, called the fundamental variety, whose coordinate ring coincides with the nilradical quotient of the complexification of degree $0$ abelian knot contact homology. Using this framework, we introduce the notion of \emph{ghost characters} and prove that Ng's conjecture holds for a knot $K$ if and only if $K$ admits no ghost characters. This criterion establishes Ng's conjecture for all 2-bridge and 3-bridge knots.