NATO Science Series
DOI: 10.1007/978-1-4020-2772-7_1
|View full text |Cite
|
Sign up to set email alerts
|

Combinatorial Formulas for Cohomology of Spaces of Knots

Abstract: Abstract. We develop homological techniques for finding explicit combinatorial expressions of finite-type cohomology classes of spaces of knots in R n , n ≥ 3, generalizing Polyak-Viro formulas [10] for invariants (i.e. 0-dimensional cohomology classes) of knots in 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 R 1 ֒→ R n not reducible to knot invariants or their natural stabilizat… Show more

Help me understand this report

Search citation statements

Order By: Relevance

Paper Sections

Select...
3
1
1

Citation Types

1
47
0
12

Publication Types

Select...
4
1
1

Relationship

1
5

Authors

Journals

citations
Cited by 16 publications
(60 citation statements)
references
References 18 publications
1
47
0
12
Order By: Relevance
“…Any invariant given by Polyak-Viro formulas has a finite degree; by the Goussarov theorem [8] the converse also is true: any finite degree invariant can be expressed by such a formula. There exist also some other combinatorial formulas for some invariants, in particular the ones described in [12], [4], and in the present work (see also [24], [27]). …”
Section: Introductionsupporting
confidence: 66%
See 2 more Smart Citations
“…Any invariant given by Polyak-Viro formulas has a finite degree; by the Goussarov theorem [8] the converse also is true: any finite degree invariant can be expressed by such a formula. There exist also some other combinatorial formulas for some invariants, in particular the ones described in [12], [4], and in the present work (see also [24], [27]). …”
Section: Introductionsupporting
confidence: 66%
“…This work is a continuation of [24] but can be read independently. In [24], a general geometrical approach to the construction of combinatorial formulas of cohomology classes of spaces of knots in R n , n ≥ 3, was proposed.…”
Section: Introductionmentioning
confidence: 99%
See 1 more Smart Citation
“…This fact was proved in Vassiliev (2001) by means of an explicit combinatorial formula (see gure 8 and the following theorem).…”
Section: (I) Example: Teiblum{turchin Cocycle and Its Realizationmentioning
confidence: 87%
“…Namely, for 'compact' knots S 1 → R n there are two linearly independent cohomology classes of filtration 1 (of dimensions n − 2 and n − 1) and two cohomology classes of filtration 2 (one of which is the well-known knot invariant or its stabilization mentioned in I and has dimension 2(n−3), and the second is of dimension 2n − 3). For 'long' knots R 1 → R n there are no cohomology classes of filtration 1 or 2 other than the knot invariant or its stabilization, and in filtration 3 for any n there is exactly one more independent cohomology class having dimension 3n − 8: it was found by D. Teiblum and V. Turchin in the case of odd n and in [69], [56] for even n. Combinatorial formulas for all these classes will be given in [74]. L. Multiplication.…”
Section: 8mentioning
confidence: 99%