Bott-Taubes-Vassiliev cohomology classes by cut-and-paste topology

TL;DR

This paper develops a method to construct integer-valued cohomology classes in knot and link spaces in higher dimensions, extending previous real-valued classes and introducing mod-p classes using graph cocycles and configuration space techniques.

Contribution

It introduces a new approach to produce integer and mod-p cohomology classes from graph cocycles in higher-dimensional knot spaces, expanding the scope of Vassiliev invariants.

Findings

Constructed integer-valued cohomology classes from graph cocycles.

Established a lattice of integer classes within real cohomology.

Extended methods to mod-p classes not reducible from integers.

Abstract

Bott and Taubes used integrals over configuration spaces to produce finite-type a.k.a. Vassiliev knot invariants. Cattaneo, Cotta-Ramusino and Longoni then used these methods together with graph cohomology to construct "Vassiliev classes" in the real cohomology of spaces of knots in higher-dimensional Euclidean spaces, as first promised by Kontsevich. Here we construct integer-valued cohomology classes in spaces of knots and links in Euclidean spaces of dimension greater than three. We construct such a class for any integer-valued graph cocycle, by the method of gluing compactified configuration spaces. Our classes form the integer lattice among the previously discovered real cohomology classes. Thus we obtain nontrivial classes from trivalent graph cocycles. Our methods generalize to yield mod-p classes out of mod-p graph cocycles, which need not be reductions of classes over the integers.