Moscow Mathematical Journal
 RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Archive Impact factor Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Mosc. Math. J.: Year: Volume: Issue: Page: Find

 Mosc. Math. J., 2001, Volume 1, Number 1, Pages 91–123 (Mi mmj14)

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.

DOI: https://doi.org/10.17323/1609-4514-2001-1-1-91-123

Full text: http://www.ams.org/.../abstracts-1-1.html
References: PDF file   HTML file

Bibliographic databases:

MSC: Primary 57M25, 55R80; Secondary 57Q45, 55T99, 54F05
Language:

Citation: V. A. Vassiliev, “Combinatorial formulas for cohomology of knot spaces”, Mosc. Math. J., 1:1 (2001), 91–123

Citation in format AMSBIB
\Bibitem{Vas01} \by V.~A.~Vassiliev \paper Combinatorial formulas for cohomology of knot spaces \jour Mosc. Math.~J. \yr 2001 \vol 1 \issue 1 \pages 91--123 \mathnet{http://mi.mathnet.ru/mmj14} \crossref{https://doi.org/10.17323/1609-4514-2001-1-1-91-123} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1852936} \zmath{https://zbmath.org/?q=an:1015.57003} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000208587300006} \elib{https://elibrary.ru/item.asp?id=6003842} 

• http://mi.mathnet.ru/eng/mmj14
• http://mi.mathnet.ru/eng/mmj/v1/i1/p91

 SHARE:

Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. V. A. Vassiliev, “Topology of plane arrangements and their complements”, Russian Math. Surveys, 56:2 (2001), 365–401
2. Vassiliev V.A., “Homology of spaces of knots in any dimensions”, Philosophical Transactions of the Royal Society of London Series A-Mathematical Physical and Engineering Sciences, 359:1784 (2001), 1343–1364
3. Johnston J.R., “Parental alignments and rejection: An empirical study of alienation in children of divorce”, Journal of the American Academy of Psychiatry and the Law, 31:2 (2003), 158–170
4. Ostlund O.P.F., “A diagrammatic approach to link invariants of finite degree”, Mathematica Scandinavica, 94:2 (2004), 295–319
5. Vassiliev V.A., “Combinatorial formulas for cohomology of spaces of knots”, Advances in Topological Quantum Field Theory, NATO Science Series, Series II: Mathematics, Physics and Chemistry, 179, 2004, 1–21
6. S. V. Alenov, “Arrow-diagram formulas for fourth-order invariants of knots”, J. Math. Sci., 146:1 (2007), 5455–5464
7. V. A. Vassiliev, “First-order invariants and cohomology of spaces of embeddings of self-intersecting curves in $\mathbb R^n$”, Izv. Math., 69:5 (2005), 865–912
8. V. É. Turchin, “What is one-term relation for higher homology of long knots”, Mosc. Math. J., 6:1 (2006), 169–194
9. V. É. Turchin, “Calculating the First Nontrivial 1-Cocycle in the Space of Long Knots”, Math. Notes, 80:1 (2006), 101–108
10. S. V. Alenov, V. P. Leksin, “On Diagram Formulas for Knot Invariants”, Proc. Steklov Inst. Math., 252 (2006), 4–11
11. Belmonte A., “The tangled web of self-tying knots”, Proceedings of the National Academy of Sciences of the United States of America, 104:44 (2007), 17243–17244
12. Sakai K., “Configuration Space Integrals for Embedding Spaces and the Haefliger Invariant”, J Knot Theory Ramifications, 19:12 (2010), 1597–1644
13. Sakai K., “An Integral Expression of the First Nontrivial One-Cocycle of the Space of Long Knots in R-3”, Pacific J Math, 250:2 (2011), 407–419