Geometry & Topology

A rational noncommutative invariant of boundary links

Stavros Garoufalidis and Andrew Kricker

Full-text: Open access

Abstract

In 1999, Rozansky conjectured the existence of a rational presentation of the Kontsevich integral of a knot. Roughly speaking, this rational presentation of the Kontsevich integral would sum formal power series into rational functions with prescribed denominators. Rozansky’s conjecture was soon proven by the second author. We begin our paper by reviewing Rozansky’s conjecture and the main ideas that lead to its proof. The natural question of extending this conjecture to links leads to the class of boundary links, and a proof of Rozansky’s conjecture in this case. A subtle issue is the fact that a ‘hair’ map which replaces beads by the exponential of hair is not 1-1. This raises the question of whether a rational invariant of boundary links exists in an appropriate space of trivalent graphs whose edges are decorated by rational functions in noncommuting variables. A main result of the paper is to construct such an invariant, using the so-called surgery view of boundary links and after developing a formal diagrammatic Gaussian integration. Since our invariant is one of many rational forms of the Kontsevich integral, one may ask if our invariant is in some sense canonical. We prove that this is indeed the case, by axiomatically characterizing our invariant as a universal finite type invariant of boundary links with respect to the null move. Finally, we discuss relations between our rational invariant and homology surgery, and give some applications to low dimensional topology.

Article information

Source
Geom. Topol., Volume 8, Number 1 (2004), 115-204.

Dates
Received: 10 June 2002
Accepted: 16 January 2004
First available in Project Euclid: 21 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513883365

Digital Object Identifier
doi:10.2140/gt.2004.8.115

Mathematical Reviews number (MathSciNet)
MR2033481

Zentralblatt MATH identifier
1075.57004

Subjects
Primary: 57N10: Topology of general 3-manifolds [See also 57Mxx]
Secondary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}

Keywords
boundary links Kontsevich integral Cohn localization

Citation

Garoufalidis, Stavros; Kricker, Andrew. A rational noncommutative invariant of boundary links. Geom. Topol. 8 (2004), no. 1, 115--204. doi:10.2140/gt.2004.8.115. https://projecteuclid.org/euclid.gt/1513883365


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