Homology of spaces of knots in any dimensions

Abstract

I shall describe the recent progress in the study of cohomology rings of spaces of knots in R n , H ¤ (fknots in R n g), with arbitrary n > 3.`Any dimensions' in the title can be read as dimensions n of spaces R n , as dimensions i of the cohomology groups H i , and also as a parameter for di¬erent generalizations of the notion of a knot.An important subproblem is the study of knot invariants. In our context, they appear as zero-dimensional cohomology classes of the space of knots in R 3 . It turns out that our more general problem is never less beautiful. In particular, nice algebraic structures arising in the related homological calculations have equally (or maybe even more) compact description, of which the classical`zero-dimensional' part can be obtained by easy factorization.There are many good expositions of the theory of related knot invariants. Therefore, I shall deal almost completely with results in higher (or arbitrary) dimensions.Keyword s: op erads; order complexes; simplicial resolutions; combinatorial formulae; sp ectral sequences Main constructionWe consider both the standard compact knots, i.e. smooth embeddings S 1 ! R n , and the long knots, i.e. embeddings R 1 ! R n coinciding with a standard linear embedding outside some compact subset in R 1 (see gure 1). The study of the latter space is more essential, because the algebraic structure of the cohomology ring of the space of standard knots is built from that of the similar ring for long knots (here playing the role of the`coe¯cient ring'), the topological non-triviality of the circle S 1 , and a certain family of its con guration spaces.Let us denote by K the space of all smooth maps S 1 ! R n (respectively, of maps R 1 ! R n with such boundary conditions). This is a linear (respectively, an a¯ne) space. The discriminant § » K is the set of all maps which are not smooth embeddings, i.e. have either self-intersections or singular points. The space of knots is the di¬erence K n § . (a) Arnold's reductionIt is convenient to study the cohomology group of the space of knots by a sort of Alexander duality,The bar in the notation · H ¤ means that we consider Borel{Moore homology, i.e. the homology group of the one point compacti cation, and n1 is the notation for

Calculating the first nontrivial 1-cocycle in the space of long knots

“…V. A. Vassiliev generalized the construction of this cocycle for spaces of long knots in the space R n of arbitrary dimension n ≥ 3, cf. [6,7], and then found its explicit description mod 2, cf. [1].…”

Исчисление Первого Нетривиального 1-Коцикла Пространства Длинных Узлов

“…Васильев обобщил этот коцикл на случай длинных узлов в пространстве R n произвольной размерности n 3, см. [6], [7], и затем построил его точное описание по mod 2, см. [1].…”

Topology of plane arrangements and their complements

“…The homological study of the arising resolution space is known as the theory of finite-type knot invariants, see [10], and different its generalizations, including the (equally interesting) calculation of higher dimensional cohomology classes of spaces of knots, see [78], [87], [71]. In Table 1 we give a short list of parallel notions and objects in both theories.…”