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Extending degree-2 Azumaya algebras with C2-actions and examples from character varieties of knot group

Justin Lawrence, Nicholas Rouse, Ben Williams

TL;DR

The paper develops criteria to determine when a degree-$2$ Azumaya algebra with a $C_2$-action, defined on a dense open subvariety $W$ of a curve $Y$, extends to all of $Y$ with the $C_2$-action. A local obstruction framework, based on Colliot-Thélène–type arguments and DVR analysis of maximal orders, reduces the problem to residue conditions at codimension-1 points. The authors apply this theory to canonical components of $ ext{SL}(2)$ character varieties arising from knot groups, introducing the tautological algebra $ ext{𝒜}$ on a dimension-1 component and expressing its restriction to the fraction field as a quaternion (symbol) algebra; they provide explicit presentations and compute extension obstructions in concrete knot-symmetric cases. In the Figure-8 knot example, three $C_2$-actions are studied: the strong inversion extends with action, while two $2$-periodic symmetries fail to extend with the action, illustrating the obstruction criteria in practice. The work combines theoretical criteria with computer-assisted computations (e.g., Magma) to illuminate how knot symmetry influences the extendability of tautological Azumaya algebras.

Abstract

We give criteria to determine when a degree-2 Azumaya algebra with $C_2$-action over a dense open subvariety of a curve extends to the entire curve as an algebra with $C_2$-action. These consist of conditions for the extension of the algebra, combined with a new condition for the extension of the algebra with the action. The new condition is testable by computer algebra systems, and we explain how the result applies to the canonical components of the character varieties of certain hyperbolic knots with order-2 symmetries. We conclude by carrying out the calculations for different symmetries of the Figure-8 knot.

Extending degree-2 Azumaya algebras with C2-actions and examples from character varieties of knot group

TL;DR

The paper develops criteria to determine when a degree- Azumaya algebra with a -action, defined on a dense open subvariety of a curve , extends to all of with the -action. A local obstruction framework, based on Colliot-Thélène–type arguments and DVR analysis of maximal orders, reduces the problem to residue conditions at codimension-1 points. The authors apply this theory to canonical components of character varieties arising from knot groups, introducing the tautological algebra on a dimension-1 component and expressing its restriction to the fraction field as a quaternion (symbol) algebra; they provide explicit presentations and compute extension obstructions in concrete knot-symmetric cases. In the Figure-8 knot example, three -actions are studied: the strong inversion extends with action, while two -periodic symmetries fail to extend with the action, illustrating the obstruction criteria in practice. The work combines theoretical criteria with computer-assisted computations (e.g., Magma) to illuminate how knot symmetry influences the extendability of tautological Azumaya algebras.

Abstract

We give criteria to determine when a degree-2 Azumaya algebra with -action over a dense open subvariety of a curve extends to the entire curve as an algebra with -action. These consist of conditions for the extension of the algebra, combined with a new condition for the extension of the algebra with the action. The new condition is testable by computer algebra systems, and we explain how the result applies to the canonical components of the character varieties of certain hyperbolic knots with order-2 symmetries. We conclude by carrying out the calculations for different symmetries of the Figure-8 knot.

Paper Structure

This paper contains 8 sections, 9 theorems, 46 equations, 3 figures.

Table of Contents

  1. The structure group
  2. The local calculation
  3. Character Varieties of Dimension 1
  4. Interpreting the tautological algebra
  5. The Algebra over the Fraction Field
  6. Examples from knot theory
  7. The Figure-$8$ Knot
  8. The $2$-periodicities

Key Result

Proposition 1.3

Let $W$ be a connected scheme on which $2$ is invertible. Let $\mathcal{A}$ be a degree-$2$ Azumaya algebra over $W$ and $g : \mathcal{A} \to \mathcal{A}$ a nontrivial order-$2$ isomorphism of $\mathcal{A}$. Locally in the étale topology, the algebra with $C_2$-action $(\mathcal{A}, g)$ is isomorphi

Figures (3)

  • Figure 1: The Figure-8 Knot, generators for the knot group, and two indicated symmetries.
  • Figure 2: The Figure-8 Knot and generators for the knot group after application of $\rho$.
  • Figure 3: The Figure-8 Knot and generators for the knot group after application of $\sigma$.

Theorems & Definitions (22)

  • Remark 1.1
  • Example 1.2
  • Proposition 1.3
  • proof
  • Proposition 2.1: Colliot-Thelene1979
  • proof
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • proof
  • ...and 12 more