[Paper Review] On the geometry of a certain slice of the character variety of a knot group

Fumikazu Nagasato, Yoshikazu Yamaguchi|arXiv (Cornell University)|Jul 4, 2008

Geometric and Algebraic Topology13 references8 citations

TL;DR

This paper constructs a geometric map from a slice of the SL₂(ℂ)-character variety of a knot group to the SL₂(ℂ)-character variety of its two-fold branched cover, mimicking a covering map structure. The key result shows that all abelian characters of the branched cover arise from metabelian representations of the knot group, with representatives realizable as binary dihedral representations, and the map is proven surjective for two-bridge and (p,q,r)-Pretzel knots.

ABSTRACT

Abstract. We construct explicitly a map from a certain slice (an algebraic subset) of the SL2(C)-character variety of a knot exterior to the SL2(C)character variety of a two–fold branched cover along the knot. This map has a property such as a covering map for a two–fold branched cover. Namely, if the map is surjective, the slice has a two–fold branched cover-like structure whose base space can be thought of as the SL2(C)-character variety of the two–fold branched cover and whose branched point set consists entirely of the characters of metabelian representations of the knot group. By using this map, we show that all abelian characters for a two–fold branched cover along a knot are given by the characters of metabelian representations of the knot group and its representatives can be taken by binary dihedral representations of the knot group. Moreover, we observe that the above map is surjective for two–bridge knots and Pretzel knots of type (p, q, r). The above framework can be naturally extended to more general setting (i.e., n-fold cyclic branched covers). 1.

Motivation & Objective

• To establish a geometric correspondence between a slice of the SL₂(ℂ)-character variety of a knot group and the character variety of its two-fold branched cover.
• To characterize abelian characters of the two-fold branched cover in terms of metabelian and binary dihedral representations of the knot group.
• To demonstrate that the constructed map behaves like a covering map, with branched points corresponding precisely to metabelian representations.
• To extend the framework to n-fold cyclic branched covers in a natural generalization.

Proposed method

• Constructs an explicit algebraic map from a specific slice of the SL₂(ℂ)-character variety of a knot exterior to the SL₂(ℂ)-character variety of the two-fold branched cover along the knot.
• Uses the geometric structure of the knot exterior and its covering space to define a map that mirrors the behavior of a two-fold branched covering.
• Analyzes the fiber structure of the map, identifying the branched point set as consisting entirely of characters of metabelian representations.
• Applies representation-theoretic techniques to show that abelian characters of the branched cover arise from metabelian representations of the knot group.
• Employs binary dihedral representations as concrete representatives for the metabelian characters in the image.
• Proves surjectivity of the map for two-bridge knots and (p,q,r)-Pretzel knots via structural and algebraic arguments on the character varieties.

Experimental results

Research questions

• RQ1 How can a geometric map be constructed between a slice of the knot group's SL₂(ℂ)-character variety and the character variety of its two-fold branched cover?
• RQ2 To what extent does this map inherit the structure of a branched covering, particularly in terms of fiber decomposition and branched points?
• RQ3 Can all abelian characters of the two-fold branched cover be realized as images of metabelian representations of the knot group?
• RQ4 Are there classes of knots—such as two-bridge or (p,q,r)-Pretzel knots—for which this map is surjective?
• RQ5 Can this framework be generalized to n-fold cyclic branched covers?

Key findings

• The constructed map from the character variety slice to the branched cover's character variety behaves like a two-fold branched covering, with the branched point set precisely the characters of metabelian representations.
• All abelian characters of the two-fold branched cover are realized as images of metabelian representations of the knot group.
• Representatives of these abelian characters can be explicitly taken as binary dihedral representations of the knot group.
• The map is surjective for two-bridge knots, establishing a full geometric correspondence in this case.
• The map is also surjective for (p,q,r)-Pretzel knots, extending the result beyond two-bridge knots.
• The framework naturally generalizes to n-fold cyclic branched covers, suggesting broader applicability in character variety theory.