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

This article is cited in 13 scientific papers (total in 13 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.


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MSC: Primary 57M25, 55R80; Secondary 57Q45, 55T99, 54F05
Received: October 10, 2000

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

Citation in format AMSBIB
\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

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    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  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    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  crossref  mathscinet  zmath  adsnasa  isi
    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  isi
    4. Ostlund O.P.F., “A diagrammatic approach to link invariants of finite degree”, Mathematica Scandinavica, 94:2 (2004), 295–319  crossref  mathscinet  zmath  isi
    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  mathscinet  zmath  isi
    6. S. V. Alenov, “Arrow-diagram formulas for fourth-order invariants of knots”, J. Math. Sci., 146:1 (2007), 5455–5464  mathnet  crossref  mathscinet  elib
    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  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    8. V. É. Turchin, “What is one-term relation for higher homology of long knots”, Mosc. Math. J., 6:1 (2006), 169–194  mathnet  crossref  mathscinet  zmath
    9. V. É. Turchin, “Calculating the First Nontrivial 1-Cocycle in the Space of Long Knots”, Math. Notes, 80:1 (2006), 101–108  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    10. S. V. Alenov, V. P. Leksin, “On Diagram Formulas for Knot Invariants”, Proc. Steklov Inst. Math., 252 (2006), 4–11  mathnet  crossref  mathscinet  elib
    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  crossref  adsnasa  isi
    12. Sakai K., “Configuration Space Integrals for Embedding Spaces and the Haefliger Invariant”, J Knot Theory Ramifications, 19:12 (2010), 1597–1644  crossref  mathscinet  zmath  isi  elib
    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  crossref  mathscinet  zmath  isi  elib
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