[Paper Review] Finite type invariants of cyclic branched covers

Stavros Garoufalidis, Andrew Kricker|ArXiv.org|Jul 30, 2001

Geometric and Algebraic Topology14 references4 citations

TL;DR

This paper provides a formula for the LMO invariant of cyclic branched covers of knots in integer homology spheres, expressing it in terms of the knot's signature function and residues of its rational Kontsevich integral. The key result generalizes earlier work on Casson-Walker invariants and establishes a precise link between finite type invariants and branched covering invariants via the rational lift of the Kontsevich integral.

ABSTRACT

Updated rerefences and introduction. Given a knot in an integer homology sphere, one can construct a family of closed 3-manifolds (parametrized by the positive integers), namely the cyclic branched coverings of the knot. In this paper we give a formula for the the Casson-Walker invariants of these 3-manifolds in terms of residues of a rational function (which measures the 2-loop part of the Kontsevich integral of a knot) and the signature function of the knot. Our main result actually computes the LMO invariant of cyclic branched covers in terms of a rational invariant of the knot and its signature function. Revised version.

Motivation & Objective

• To derive a general formula for the LMO invariant of cyclic branched covers of knots in integer homology spheres.
• To resolve the long-standing problem of computing the Casson-Walker invariant of such covers for all p > 2.
• To establish a precise connection between the 2-loop part of the rational Kontsevich integral and topological invariants of branched covers.
• To formalize the role of the signature function and rational invariants in the context of finite type invariants of 3-manifolds.

Proposed method

• The authors use the rational lift $ Z^{\mathrm{rat}} $ of the Kontsevich integral, constructed via perturbative field theory and the Aarhus integral.
• They apply the twisting map $ \tau^{\mathrm{rat}} $ and lifting map $ \mathrm{Lift}_p $ to relate the rational invariant to the LMO invariant of the branched cover.
• The method relies on formal calculations involving integration over diagrams with beads, using function-theoretic properties of $ Z^{\mathrm{rat}} $.
• Key operations include 'completing the square' in the context of wheel diagrams and integration by parts on the rational integral.
• The approach leverages the universal abelian cover's structure, which maps onto all cyclic branched covers, explaining the appearance of $ Z^{\mathrm{rat}} $.
• The formula incorporates framing corrections via the signature term $ \sigma_p(M,K)\Theta/16 $, accounting for normalization differences.

Experimental results

Research questions

• RQ1 How can the LMO invariant of a cyclic branched cover of a knot be expressed in terms of knot invariants?
• RQ2 What is the precise role of the 2-loop part of the rational Kontsevich integral in computing invariants of branched covers?
• RQ3 Why does the signature function appear in the formula for the Casson-Walker invariant of branched covers?
• RQ4 How do the twisting and lifting operations on diagrams relate to the geometry of cyclic branched covers?
• RQ5 Can the rational lift $ Z^{\mathrm{rat}} $ be used to compute finite type invariants of 3-manifolds arising from knot constructions?

Key findings

• The LMO invariant of the $ p $-fold cyclic branched cover $ \Sigma^p_{(M,K)} $ is given by $ Z(\Sigma^p_{(M,K)}) = e^{\sigma_p(M,K)\Theta/16} \cdot \mathrm{Lift}_p \circ \tau^{\mathrm{rat}}_{\alpha_p} \circ Z^{\mathrm{rat}}(M,K) $, where $ \alpha_p = \nu^{-(p-1)/p} $.
• The formula holds for all $ p $-regular knots, i.e., those for which the branched cover is a rational homology sphere.
• The Casson-Walker invariant of the branched cover is determined by the signature function $ \sigma_p(M,K) $ and the residues of the 2-loop part of $ Z^{\mathrm{rat}}(M,K) $.
• The rational invariant $ Z^{\mathrm{rat}} $ captures the 2-loop contribution to the Kontsevich integral and is essential for computing the invariant in the presence of framing anomalies.
• The derivation uses formal manipulations involving integration over diagrams, completing the square, and the integration-by-parts lemma for rational integrals.
• The result confirms a conjecture from [GR] and extends earlier results for $ p=2 $ to all $ p \geq 2 $, providing a unified framework for finite type invariants of branched covers.