Russian:  ps

Russian:  ps

English:  ps (gzipped)

Abstract. An alternative link representation different from planar diagrams is discussed. Isotopy classes of unordered nonoriented links are realized as central elements of a monoid presented explicitly by a finite number of generators and relations. A new partial algorithm for recognizing is constructed. Experiments show that the algorithm allows to untangle unknots whose planar diagram has hundreds of crossings. Here `to untangle' means `to find an isotopy to the circle'

Russian:  ps

Abstract. In this paper we construct finitely presented semigroups whose central elements are in one-to-one correspondence with isotopy classes of non-oriented links in three-space. Solving the word problem for those semigroups is equivalent to solving the classification problem for links and tangles. Also, we give a construction of finitely presented groups containing the braid group as a subgroup.

Russian:  ps (gzipped)
English:  ps

Russian:  ps
English:  ps

English:  ps (gzipped)

Abstract. Let $\GR \to B$ be a bundle of compact Lie groups acting on a fiber bundle $Y \to B$. In this paper we introduce and study gauge-equivariant $K$-theory groups $K_\GR^i(Y)$. These groups satisfy the usual properties of the equivariant $K$-theory groups, but also some new phenomena arise due to the topological non-triviality of the bundle $\GR \to B$. As an application, we define a gauge-equivariant index for a family of elliptic operators $(P_b)_{b \in B}$ invariant with respect to the action of $\GR \to B$, which, in this approach, is an element of $K_\GR^0(B)$. We then give another definition of the gauge-equivariant index as an element of $K_0(C^*(\GR))$, the $K$-theory group of the Banach algebra $C^*(\GR)$. We prove that $K_0(C^*(\GR))\simeq K^0_\GR(\GR)$ and that the two definitions of the gauge-equivariant index are equivalent. The algebra $C^*(\GR)$ is the algebra of continuous sections of a certain field of $C^*$-algebras with non-trivial Dixmier-Douady invariant. The gauge-equivariant $K$-theory groups are thus examples of twisted $K$-theory groups, which have recently proved themselves useful in the study of Ramond-Ramond fields.

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Abstract. This work is dedicated to some new exotic homological constructions associated with the different Morse-type inequalities for differential forms and vector fields. It contains also survey of ideas developed by the present author in 1986 for this goal.