On combinatorial formulas for cohomology of spaces of knots

2014/07/27 by V. A. Vassiliev, Victor A. Vassiliev, Vassiliev, Victor A.
Mathematics · #Advanced Combinatorial Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #math.GT

paper · pdf · doi:10.48550/arxiv.1407.7235

arxiv created 2014/07/27 · arxiv updated 2014/07/29

Abstract

We develop homological techniques for finding explicit combinatorial expressions of finite-type cohomology classes of spaces of knots in Rⁿ, n ≥ 3, generalizing Polyak--Viro formulas for invariants (i.e. 0-dimensional cohomology classes) of knots in R³. As the first applications we give such formulas for the (reduced mod 2) \em generalized Teiblum--Turchin cocycle of order 3 (which is the simplest cohomology class of \em long knots R¹ \hookrightarrow Rⁿ not reducible to knot invariants or their natural stabilizations), and for all integral cohomology classes of orders 1 and 2 of spaces of \em compact knots S¹ \hookrightarrow Rⁿ. As a corollary, we prove the nontriviality of all these cohomology classes in spaces of knots in R³.

Related

• Topology of two-connected graphs and homology of spaces of knots
• Cohomology of spaces of complex knots
• Calculus of the first non-trivial 1-cocycle of the space of long knots
• A Geometric Homology Representative in the Space of Long Knots
• On the homology of the spaces of long knots