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This article is cited in 19 scientific papers (total in 19 papers)
Combinatorial formulas for cohomology of knot spaces
V. A. Vassiliev Independent University of Moscow
Abstract:
We develop homological techniques for finding explicit combinatorial formulas for finite-type cohomology classes of spaces of knots in $\mathbb R^n$, $n\ge 3$, generalizing the Polyak–Viro formulas [PV] for invariants (i.e., 0-dimensional cohomology classes) of knots in $\mathbb{R}^3$. As the first applications, we give such formulas for the (reduced mod 2) generalized Teiblum-Turchin cocycle of order 3 (which is the simplest cohomology class of long knots $\mathbb R^1\hookrightarrow\mathbb R^n$ not reducible to knot invariants or their natural stabilizations), and for all integral cohomology classes of orders 1 and 2 of spaces of compact knots $S^1\hookrightarrow\mathbb R^n$. As a corollary, we prove the nontriviality of all these cohomology classes in spaces of knots in $\mathbb R^3$.
Key words and phrases:
Knot theory, discriminant, combinatorial formula, simplicial resolution, spectral sequence, chord diagram, finite-type cohomology class.
Received: October 10, 2000
Citation:
V. A. Vassiliev, “Combinatorial formulas for cohomology of knot spaces”, Mosc. Math. J., 1:1 (2001), 91–123
Linking options:
https://www.mathnet.ru/eng/mmj14 https://www.mathnet.ru/eng/mmj/v1/i1/p91
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