Trace-free characters and abelian knot contact homology I

Abstract: This research investigates the mechanism underlying Ng's conjecture on the relationship between degree 0 abelian knot contact homology and the character variety of the 2-fold branched cover of the 3-sphere along a knot. The key idea involves analyzing trace-free characters of knot groups, which form the cross-section of $\SL_2(\C)$-character variety of a knot group cut by the hyperplane defined by the trace-free condition on meridians. This cross-section is referred to as the trace-free slice of the character variety (or simply the trace-free slice of a knot). We present a set of equations derived from a knot diagram whose common solutions coincide with the trace-free slice. This formulation describes the structure of the trace-free slice as a 2-fold branched cover over a closed algebraic set, referred to as the fundamental variety, and provides a powerful framework for analyzing the relationship stated in Ng's conjecture. Utilizing this framework, we introduce the notion of ghost characters of a knot. Consequently, using ghost characters, we show that Ng's conjecture holds for a knot $K$ if and only if $K$ admits no ghost characters.