Algebraic & Geometric Topology

The Milnor triple linking number of string links by cut-and-paste topology

Robin Koytcheff

Full-text: Open access

Abstract

Bott and Taubes constructed knot invariants by integrating differential forms along the fiber of a bundle over the space of knots, generalizing the Gauss linking integral. Their techniques were later used to construct real cohomology classes in spaces of knots and links in higher-dimensional Euclidean spaces. In previous work, we constructed cohomology classes in knot spaces with arbitrary coefficients by integrating via a Pontrjagin–Thom construction. We carry out a similar construction over the space of string links, but with a refinement in which configuration spaces are glued together according to the combinatorics of weight systems. This gluing is somewhat similar to work of Kuperberg and Thurston. We use a formula of Mellor for weight systems of Milnor invariants, and we thus recover the Milnor triple linking number for string links, which is in some sense the simplest interesting example of a class obtained by this gluing refinement of our previous methods. Along the way, we find a description of this triple linking number as a degree of a map from the 6–sphere to a quotient of the product of three 2–spheres.

Article information

Source
Algebr. Geom. Topol., Volume 14, Number 2 (2014), 1205-1247.

Dates
Received: 7 March 2013
Revised: 30 August 2013
Accepted: 20 October 2013
First available in Project Euclid: 19 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513715862

Digital Object Identifier
doi:10.2140/agt.2014.14.1205

Mathematical Reviews number (MathSciNet)
MR3180832

Zentralblatt MATH identifier
1297.55016

Subjects
Primary: 55R80: Discriminantal varieties, configuration spaces 57M27: Invariants of knots and 3-manifolds
Secondary: 55R12: Transfer 57R40: Embeddings

Keywords
triple linking number configuration space integrals Pontrjagin–Thom construction gluing degree string links finite-type link invariants

Citation

Koytcheff, Robin. The Milnor triple linking number of string links by cut-and-paste topology. Algebr. Geom. Topol. 14 (2014), no. 2, 1205--1247. doi:10.2140/agt.2014.14.1205. https://projecteuclid.org/euclid.agt/1513715862


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