A rational noncommutative invariant of boundary links

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.

Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol8/paper4.abs.html

References in corpus (5)

• Calculus of clovers and finite type invariants of 3-manifolds
• Two applications of elementary knot theory to Lie algebras and Vassiliev invariants
• Homology surgery and invariants of 3-manifolds
• Concordance and 1-loop clovers
• Caracteres sur l'algebre de diagrammes trivalents Lambda

Cited by in corpus (16)

• Asymptotics of the colored Jones function of a knot
• Concordance and 1-loop clovers
• Non injectivity of the "hair" map
• Whitehead doubling persists
• On Alexander modules and Blanchfield forms of null-homologous knots in rational homology spheres
• Non-commutative Chern-Weil Theory and the Combinatorics of Wheeling
• Surgery presentations of coloured knots and of their covering links
• Surgery untying of coloured knots
• A computation of the Kontsevich integral of torus knots
• Resurgence in complex Chern-Simons theory at generic levels
• A non-commutative Reidemeister-Turaev torsion of homology cylinders
• Finite type invariants of knots in homology 3-spheres with respect to null LP-surgeries
• Equivariant triple intersections
• On Bar-Natan - van der Veen's perturbed Gaussians
• Splitting formulas for the rational lift of the Kontsevich integral
• Differential Operators and the Wheels Power Series