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

TL;DR

This paper establishes criteria for extending degree-2 Azumaya algebras with C2-actions over curves, applying the results to character varieties of hyperbolic knots, and demonstrates the approach with the Figure-8 knot.

Contribution

It introduces a new testable condition for extending Azumaya algebras with C2-actions, linking algebraic criteria with geometric applications to knot character varieties.

Findings

Criteria for extending Azumaya algebras with C2-actions.

Application to character varieties of hyperbolic knots.

Explicit calculations for the Figure-8 knot.

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.