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Trace-free characters and abelian knot contact homology I

Fumikazu Nagasato

TL;DR

The paper develops a geometric framework around trace-free $SL_2(\\mathbb{C})$-characters of knot groups to illuminate Ng's conjecture linking $HC_0^{ab}(K)$ with the coordinate ring of the $2$-fold branched cover. By isolating the trace-free slice $S_0(K)$ and introducing the fundamental variety $F_2(K)$, it shows $S_0(K)$ is a $2$-fold cover of $F_2(K)$, with $\,\mathbb{C}[F_2(K)]\cong (HC_0^{ab}(K)\otimes \\mathbb{C})/\\sqrt{0}$. The work formalizes the conjecture in terms of the surjectivity of maps between these spaces, and identifies ghost characters as the precise obstruction to Ng's conjecture, proving it for all $2$-bridge and $3$-bridge knots and establishing a necessary-and-sufficient criterion for the conjecture in general. The framework clarifies how abelian knot contact homology encodes the $SL_2(\\mathbb{C})$-character geometry of the $2$-fold branched cover, with potential to guide explicit computations for broader knot families.

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 $X(Σ_2 K)$ 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 $X(Σ_2K)$. 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.

Trace-free characters and abelian knot contact homology I

TL;DR

The paper develops a geometric framework around trace-free -characters of knot groups to illuminate Ng's conjecture linking with the coordinate ring of the -fold branched cover. By isolating the trace-free slice and introducing the fundamental variety , it shows is a -fold cover of , with . The work formalizes the conjecture in terms of the surjectivity of maps between these spaces, and identifies ghost characters as the precise obstruction to Ng's conjecture, proving it for all -bridge and -bridge knots and establishing a necessary-and-sufficient criterion for the conjecture in general. The framework clarifies how abelian knot contact homology encodes the -character geometry of the -fold branched cover, with potential to guide explicit computations for broader knot families.

Abstract

We study the structure underlying Ng's conjecture, which relates the degree abelian knot contact homology of a knot to the coordinate ring of the -character variety of the -fold branched cover of the -sphere branched along . Our approach is based on the study of (meridionally) trace-free characters of knot groups. For each knot , they form a closed algebraic subset of the -character variety of , defined by the trace-free condition on meridians. The subset , called the trace-free slice of , has a natural connection to . We show that the trace-free slice admits the structure of a -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 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 if and only if admits no ghost characters. This criterion establishes Ng's conjecture for all 2-bridge and 3-bridge knots.

Paper Structure

This paper contains 13 sections, 11 theorems, 141 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Trace-free slice of a knot
  3. Character variety of a finitely presented group
  4. Trace-free slice of a knot
  5. Proof of Theorem \ref{['defpoly_S0K']}
  6. Character varieties of free groups and their trace-free slices
  7. Proof of Theorem \ref{['defpoly_S0K']}
  8. Trace-free slice and fundamental variety
  9. Application of the trace-free slice to abelian knot contact homology
  10. Degree 0 abelian knot contact homology
  11. Fundamental variety and degree 0 abelian knot contact homology
  12. Ng's conjecture and ghost characters of a knot
  13. Ghost characters and obstructions to Conjecture \ref{['conj_nag']}

Key Result

Theorem 2.1

Let $G(K)=\langle m_1,\cdots,m_n \mid r_1,\cdots,r_{n-1} \rangle$ be a Wirtinger presentation. Then the trace-free slice $S_0(K)$ is isomorphic to the algebraic set: where the defining equations are given as follows: with the convention $x_{ii}=2$, $x_{ji}=x_{ij}$ and $x_{i_{\sigma(1)}i_{\sigma(2)}i_{\sigma(3)}}=\mathrm{sign}(\sigma)x_{i_1i_2i_3}$ for any permutation $\sigma \in \mathfrak{S}_3$.

Figures (6)

  • Figure 1: Decomposition of the exterior $E_K$ for the figure-eight knot. The attaching curves of the $2$-handles are shown on the boundary of the handlebody $H_4$. The meridians of $K$ are right-handed, with orientations indicated by arrows on the tubes.
  • Figure 2: Skein $s_{i_1 \dots i_k}$ for $1 \leq i_1, \cdots, i_k \leq n$ in the handlebody $H_n \cong D_n \times [0,1]$, regarded as a planar curve on $D_n \times \{0\}$. Every meridian $m_i$ is oriented clockwise.
  • Figure 3: Attaching curves and chords. Each of the cross signs "$\times$" on chords presents a crossing of the attaching curve where the sign is.
  • Figure 4: Intersetion patterns of the loop representing $m_im_j$ with the surface $S \cap E_K$. $N_K$ denotes a tubular neighborhood of $K$.
  • Figure 5: Schematic view of $X(\Sigma_2K)$, $S_0(K)$ and $F_2(K)$ with a ghost character.
  • ...and 1 more figures

Theorems & Definitions (18)

  • Theorem 2.1: cf. Nagasato4, Theorem 3.2 in Gonzalez-Montesinos
  • Proposition 3.1: cf. Gonzalez-Montesinos
  • Theorem 3.2: BullockPrzytycki-Sikora
  • Theorem 3.3: Przytycki1
  • Lemma 3.4
  • Definition 4.1: degree 0 abelian knot contact homology
  • Proposition 4.2: cf. Theorem 7.5 in Nagasato3
  • Conjecture 4.3: Conjecture 5.7 in Ng2
  • Conjecture 4.4
  • Theorem 4.5: Fox, cf.Kinoshita
  • ...and 8 more