Denis Petrovich Ilyutko

Virtual knots: the state of the art

Basic Definitions and Notions Virtual Knots and Three-Dimensional Topology Quandles (Distributive Groupoids) in Virtual Knot Theory The Jones Polynomial. Atoms Khovanov Homology Virtual Braids Vassilievs Invariants Parity in Knot Theory. Free-Knots. Cobordisms Theory of Graph-Links.

AN EQUIVALENCE BETWEEN THE SET OF GRAPH-KNOTS AND THE SET OF HOMOTOPY CLASSES OF LOOPED GRAPHS

In the present paper we construct a one-to-one correspondence between the set of graph-knots and the set of homotopy classes of looped graphs. Moreover, the graph-knot and the homotopy class constructed from a given knot are related with this correspondence. This correspondence is given by a simple formula.

Разветвленные экстремали функционала

В статье рассматриваются сети на λ-нормированных плоскостях, т.е. на нормированных плоскостях, для которых единичная окружность является правильным 2λ-угольником. Дается геометрический критерий экстремальности произвольного дерева на λ-нормированной плоскости, где λ 6= 2, 3, 4, 6. Затрагиваются также вопросы о λ-минимальной (экстремальной) реализации произвольной сети и о сходимости λ-экстремальных сетей при λ→∞. Библиография: 17 названий.

lambda

It is well known that there are many presentations of a knot and each presentation has both its own advantages and disadvantages. In the paper we present another presentation of a knot. For the new presentation we define moves and rewrite the criterion of realizability.

-нормированной длины@@@Branching extremals of the functional of

For some class of mappings, there are investigated problems connected with a possibility of continuous extension to a boundary on Riemannian manifolds. In particular, for so-called ring mappings, there is proved a result related to continuous extension to an isolated boundary point. Besides that, similar theorems hold for more general boundaries. As application of developed technique, there is proves a theorem about continuous extension of mapping of Orlicz--Sobolev class to an isolated singularity.

lambda

The paper is a survey of a series of publications of the authors awarded by I. I Shuvalov First Prize of Lomonosov Moscow State University for scientific activity, 2017.