[Paper Review] On the geometry of the slice of trace--free SL(2,C)-characters of a knot group

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

Geometric and Algebraic Topology17 references4 citations

TL;DR

This paper constructs a map Φ from the trace-free SL(2,C)-character variety of a knot exterior to the character variety of its 2-fold branched cover, showing that this map is surjective for 2-bridge and certain pretzel knots. The key result is that the slice of trace-free characters is a 2-fold branched cover over the character variety of the 2-fold branched cover, with branched locus precisely the metabelian characters, which correspond to binary dihedral representations.

ABSTRACT

Let K be a knot in an integral homology 3-sphere and let B denote the 2-fold branched cover of the integral homology sphere branched along K. We construct a map from the slice of characters with trace free along meridians in the SL(2, C)-character variety of the knot exterior to the SL(2, C)-character variety of 2-fold branched cover B. When this map is surjective, it describes the slice as the 2-fold branched cover over the SL(2, C)-character variety of B with branched locus given by the abelian characters, whose preimage is precisely the set of metabelian characters. We show that each of metabelian character can be represented as the character of a binary dihedral representation of the knot group. This map is shown to be surjective for all 2-bridge knots and all pretzel knots of type (p, q, r). An extension of this framework to n-fold branched covers is also described.

Motivation & Objective

• To understand the relationship between SL(2,C)-representations of knot exteriors and their finite cyclic covers, particularly 2-fold branched covers.
• To define and analyze a map Φ from the trace-free character slice of a knot exterior to the character variety of its 2-fold branched cover.
• To characterize the image of this map and identify the branched locus as the set of abelian characters, whose preimage consists of metabelian characters.
• To show that metabelian characters arise from binary dihedral representations of the knot group.
• To establish surjectivity of the induced map on characters for 2-bridge and (p,q,r)-pretzel knots.

Proposed method

• Define a map Φ from SL(2,C)-representations of the knot group π₁(E_K) with trace-zero meridians to representations of π₁(B₂), the fundamental group of the 2-fold branched cover B₂.
• Construct the induced map  Φ̂ on character varieties, restricting to the slice S₀(E_K) of trace-free characters.
• Use the presentation of π₁(B₂) for 2-bridge and pretzel knots, particularly the Brieskorn manifold structure for pretzel knots.
• Analyze the action of the covering transformation τ on π₁(B₂) and show that irreducible representations are τ-equivariant, implying τ*-invariance.
• Prove that the image of  Φ̂ is the τ-invariant part of X(B₂), and show that all abelian and irreducible characters in X(B₂) lie in the image.
• Use conjugation by the matrix k = [[0,√-1],[√-1,0]] to verify τ-equivariance of irreducible representations.

Experimental results

Research questions

• RQ1 How does the character variety of a knot exterior with trace-zero meridians relate to the character variety of its 2-fold branched cover?
• RQ2 What is the geometric structure of the slice of trace-free SL(2,C)-characters of a knot group?
• RQ3 Which characters in the 2-fold branched cover arise as images of metabelian representations from the knot exterior?
• RQ4 Under what conditions is the map Φ̂ from the trace-free character slice to the character variety of the 2-fold branched cover surjective?
• RQ5 Can metabelian characters of a knot group be realized as characters of binary dihedral representations?

Key findings

• The map Φ̂ from the trace-free character slice S₀(E_K) to X(B₂) is surjective for all 2-bridge knots.
• For all pretzel knots of type (p,q,r), the map Φ̂ is surjective, due to the Brieskorn manifold structure of B₂.
• The slice S₀(E_K) is a 2-fold branched cover over X(B₂), with the branched locus precisely the set of abelian characters.
• The preimage of the abelian characters under Φ̂ is exactly the set of metabelian characters of the knot group.
• Each metabelian character arises as the character of a binary dihedral representation of π₁(E_K).
• Irreducible representations of π₁(B₂) are τ-equivariant, which implies that X_irr(B₂) is contained in the image of Φ̂.