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Knots with large character varieties

Philip Choi, Joan Porti, Seokbeom Yoon

TL;DR

The paper investigates knots whose $\mathrm{SL}_2(\mathbb{C})$ character varieties have high-dimensional components (called $\mathcal X$-large). It develops two diagrammatic methods—rational tangle replacements on split-link diagrams and braid-based constructions using orientation-reversing involutions—to produce $\mathcal X$-large knots, and it proves a non-orientable analogue of Thurston's dimension bound to control these varieties. The authors confirm $\mathcal X$-largeness for knots such as $10_{98}$, $10_{99}$, and $10_{123}$ and propose that all Turk's head knots $Th(p,q)$ with odd $p,q$ are $\mathcal X$-large, supported by a general dimension-estimate framework. Overall, the work connects diagrammatic knot operations to representation-theoretic properties of knot groups and introduces a non-orientable expansion of Thurston-type dimension bounds that may influence broader studies of $\mathrm{SL}_2(\mathbb C)$-character varieties.

Abstract

We study knots whose $\mathrm{SL}_2(\mathbb{C})$-character varieties have a component of dimension greater than one. We call such knots $\mathcal{X}$-large and introduce two diagrammatic constructions that produce $\mathcal{X}$-large knots. The first construction uses split link diagrams and rational tangle replacements, providing a topological explanation for most $\mathcal{X}$-large knots observed in knot tables. The second construction is based on braids and orientation-reversing involutions, and is motivated by a detailed analysis of the knot $10_{123}$, also known as the Turk's head knot $Th(3,5)$. In particular, this approach applies to Turk's head knots $Th(p,q)$ with $p$ and $q$ odd, leading us to conjecture that all such knots are $\mathcal{X}$-large. In doing so, we also present a non-orientable analogue of Thurston's theorem giving a lower bound on the dimension of character varieties of non-orientable 3-manifolds.

Knots with large character varieties

TL;DR

The paper investigates knots whose character varieties have high-dimensional components (called -large). It develops two diagrammatic methods—rational tangle replacements on split-link diagrams and braid-based constructions using orientation-reversing involutions—to produce -large knots, and it proves a non-orientable analogue of Thurston's dimension bound to control these varieties. The authors confirm -largeness for knots such as , , and and propose that all Turk's head knots with odd are -large, supported by a general dimension-estimate framework. Overall, the work connects diagrammatic knot operations to representation-theoretic properties of knot groups and introduces a non-orientable expansion of Thurston-type dimension bounds that may influence broader studies of -character varieties.

Abstract

We study knots whose -character varieties have a component of dimension greater than one. We call such knots -large and introduce two diagrammatic constructions that produce -large knots. The first construction uses split link diagrams and rational tangle replacements, providing a topological explanation for most -large knots observed in knot tables. The second construction is based on braids and orientation-reversing involutions, and is motivated by a detailed analysis of the knot , also known as the Turk's head knot . In particular, this approach applies to Turk's head knots with and odd, leading us to conjecture that all such knots are -large. In doing so, we also present a non-orientable analogue of Thurston's theorem giving a lower bound on the dimension of character varieties of non-orientable 3-manifolds.
Paper Structure (12 sections, 12 theorems, 42 equations, 9 figures)

This paper contains 12 sections, 12 theorems, 42 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Rational tangle replacements
  3. Toy example: the knot $10_{98}$
  4. General construction
  5. Braids and orientation-reversing involutions
  6. Toy example: the knot $10_{123}$
  7. General construction
  8. Dimension estimate à la Thurston
  9. Klein bottle
  10. Dimension estimate
  11. Proof of Theorem \ref{['thm.sym']}
  12. Parabolic representations

Key Result

Theorem 2.1

Let $L$ be a split link consisting of a non-trivial knot $T$ and an unknot. Fix a diagram of $L$ such that there is a crossing $c$ at which $T$ intersects with the unknot. Suppose Then $K$ is $\mathcal{X}$-large.

Figures (9)

  • Figure 1: The split link $3_1 \sqcup O$ and the knot $10_{98}$.
  • Figure 2: Replacement of a crossing by a 2-tangle.
  • Figure 3: Replacement of a crossing by a 2-tangle.
  • Figure 4: The $c$-closure of a rational 2-tangle $R$.
  • Figure 5: Examples of $\mathcal{X}$-large knots.
  • ...and 4 more figures

Theorems & Definitions (22)

  • Theorem 2.1
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Remark 3.3
  • Lemma 3.4
  • proof
  • Theorem 3.5
  • ...and 12 more