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Trace-free characters and abelian knot contact homology II

Fumikazu Nagasato, Shinnosuke Suzuki

TL;DR

This work tests Ng's conjecture linking degree $0$ abelian knot contact homology to the $2$-fold branched-cover character variety by focusing on ghost characters, obstructions arising from the trace-free slice $S_0(K)$ and the fundamental variety $F_2(K)$. The authors compute ghost characters for the $(4,5)$- and $(5,6)$-torus knots, showing they admit ghosts and thus provide counterexamples to the conjecture; they analyze how the central maps $h^*$ and $\widehat{\Phi}$ fail in these cases, using explicit braid-based elimination and $SL_2(\mathbb{C})$ representations of the branched-cover groups. The results establish that the conjectured isomorphism breaks down for these torus knots, revealing distinct obstruction mechanisms: surjectivity can fail (as for $T_{5,6}$) or injectivity can fail (as for $T_{4,5}$), and that $\widehat{\Phi}$ need not be surjective. Overall, the paper clarifies the limitations of Ng's conjecture beyond low-bridge knots and characterizes the obstruction via ghost characters, the trace-free slice, and the fundamental variety.

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.

Trace-free characters and abelian knot contact homology II

TL;DR

This work tests Ng's conjecture linking degree abelian knot contact homology to the -fold branched-cover character variety by focusing on ghost characters, obstructions arising from the trace-free slice and the fundamental variety . The authors compute ghost characters for the - and -torus knots, showing they admit ghosts and thus provide counterexamples to the conjecture; they analyze how the central maps and fail in these cases, using explicit braid-based elimination and representations of the branched-cover groups. The results establish that the conjectured isomorphism breaks down for these torus knots, revealing distinct obstruction mechanisms: surjectivity can fail (as for ) or injectivity can fail (as for ), and that need not be surjective. Overall, the paper clarifies the limitations of Ng's conjecture beyond low-bridge knots and characterizes the obstruction via ghost characters, the trace-free slice, and the fundamental variety.

Abstract

We show that the - and -torus knots admit ghost characters. Consequently, these knots provide counterexamples to Ng's conjecture, which proposes an isomorphism between the complexification of degree abelian knot contact homology and the coordinate ring of the character variety of the -fold branched cover of the -sphere branched along a knot. While Ng's conjecture has been verified for all -bridge and -bridge knots, we demonstrate, via ghost characters, how this isomorphism fails for these torus knots.

Paper Structure

This paper contains 8 sections, 7 theorems, 109 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Overview of Ng's conjecture
  3. Ghost characters of a knot and Conjecture \ref{['conj_nag']}
  4. Ghost characters of the (4,5)-torus knot
  5. Ghost characters of $T_{5,6}$
  6. Behavior of the map $h^*$
  7. Character variety $X(\Sigma_2T_{4,5})$ and behavior of $h^*$
  8. Character variety $X(\Sigma_2T_{5,6})$ and behavior of $h^*$

Key Result

Theorem 2.3

The map $h^*$ is surjective but not injective for $T_{4,5}$, and neither surjective nor injective for $T_{5,6}$. In particular, Conjecture conj_nag (Conjecture conj_ng with the nilradical quotient) does not hold for $T_{4,5}$ and $T_{5,6}$.

Figures (2)

  • Figure 3.1: A diagram of $T_{4,5}$ in braid (bridge) position and meridians $m_1,\cdots,m_{15}$. The four parallel curves connecting the both sides of the diagram are omitted.
  • Figure 3.2: A diagram of $T_{5,6}$ in braid (bridge) position and meridians $m_1,\cdots,m_{24}$. The five parallel curves connecting the both sides of the diagram are omitted.

Theorems & Definitions (11)

  • Conjecture 2.1: Conjecture 5.7 in Ng2
  • Conjecture 2.2: Conjecture 4.4 in Nagasato5
  • Theorem 2.3: cf. Theorems \ref{['thm_main2']} and \ref{['thm_main']}
  • Proposition 3.1
  • Proposition 3.2
  • Theorem 4.1
  • proof
  • Lemma 4.2
  • Theorem 4.3
  • proof
  • ...and 1 more