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 $S{L}_2(\mathbf{C})$-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 $S{L}_2(\mathbf{C})$-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 $X({\Sigma }_{2}K)$. 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.

32 pages. This paper explains and updates the results given in the talk by the author at the conference "RIMS Seminar, Representation spaces, twisted topological invariants and geometric structures of 3-manifolds" held in 2012