I. M. Nikonov

On braids and groups Gnk

In [V.O. Manturov, Non-reidemeister knot theory and its applications in dynamical systems, geometry, and topology, arxiv:1501.05208] the first named author gave the definition of

HOMOTOPICAL KHOVANOV HOMOLOGY

k

The Diagram Approach in Knot Theory and Applications to Graph Theory

-free braid groups

Height atoms whose symmetry groups act transitively on their vertex sets

G_n^k

Maximally symmetric height atoms

. Here we establish connections between free braid groups, classical braid groups and free groups: we describe explicitly the homomorphism from (pure) braid group to

Parity in knot theory and graph-links

k

Virtual knot invariants arising from parities

-free braid groups for important cases

Weak Parities and Functorial Maps

k=3,4

The Length of an Extremal Network in a Normed Space: Maxwell Formula

. On the other hand, we construct a homomorphism from (a subgroup of) free braid groups to free groups. The relations established would allow one to construct new invariants of braids and to define new powerful and easily calculated complexities for classical braid groups.

Структура хопф-циклических (ко)гомологий алгебр гладких функций@@@The Structure of the Hopf Cyclic (Co)Homology of Algebras of Smooth Functions

We modify the definition of the Khovanov complex for oriented links in a thickening of an oriented surface to obtain a triply graded homological link invariant with a new homotopical grading.