### Seminars

### on knot theory and related topics

01 February 2021, 18:30 - (UTC+3)

Title: Invariants of free knots valued in free groups

Speaker: V.O. Manturov

Organization : Knots and representation theory (by Zoom)

Abstract:

This talk is a part of the project of creating ``non-commutative'' invariants in topology. The main idea is to replace ``characteristic classes'' of moduli spaces with ``characteristic loops''. We discuss ``the last stage'' of the talk devoted to the abstract objects we get in the end: the free knots, an discuss their invariants valued in free groups.

These invariants allow one to detect easily mutations, invertibility, and other phenomena.

05 April 2021, 18:30 - (UTC+3)

Title：Virtual knot theory methods in classical topology. Part 2: Classical links on the cylinder

Organization: Knots and representation theory

Abstract: This talk is a part of the project of creating ``non-commutative'' invariants in topology. The main idea is to replace ``characteristic classes'' of moduli spaces with ``characteristic loops''. We discuss ``the last stage'' of the talk devoted to the abstract objects we get in the end: the free knots, and discuss their invariants valued in free groups. These invariants allow one to detect easily mutations, invertibility, and other phenomena. We shall use parity on the cylinder to construct classical link invariants valued in free groups.

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# Virtual knot theory methods in classical topology II : Knots in the full torus

Date: 11th May 2021

Knot theory seminar

of Roger Fenn and Louis Kauffman

Virtual knot theory methods in topology II: Partiy and cobordisms in the cylinder.

Date: 2nd June 2021

Moscow-Beijing Topology Seminar

We shall present a general theory of constructing free-group valued invariants of knots in the full torus. We shall touch on the applications of these invariants: sliceness obstructions, potential invariants of 3-manifolds

Spatial self-interlocking structures: three-dimensional and two-dimensional

Date: 4th October 2021

Knots and representation theory

The talk is devoted to the theory of self-interlocking structures and to the recent breakthrough in it:

a) There exist two-dimensional self-interlocking structures in 3-dimensional space;

b) One can construct self-interlocking 2-dimensional structures which are rigid once two polygons are fixed.

The main idea of this construction is the ``decahedron'': collection of ten (stretched) faces of the dodecahedron (without two opposite ones). They were suggested by V.O.Manturov and realised in coordinates by V.O.Manturov and S.Kim

Until recently many constructions of 3-d self-interlocking structures were known; these structures are rigid if we fix all polytopes "along the boundary". The main ideas (truncated cubes, octahedra, dodecahedra) belong to A.Ya-Belov.

The self-interlocking structure theory has many applications to architecture, composite materials, cheramics, armors etc.

The principal novelty by V.O.Manturov is the possibility of constructing these structures of "infinitely thin" polyhedra (polygons).

Surprisingly, the system of self-interlocking cubes was found by A.Ya.Belov only in 2002. It is related, in particular, with the lack of human intuition about 3-space.

In the end of the talk we'll suggest a list of problems, both purely mathematical and those related to applications.

Intersection formulas for parities on virtual knots

Igor Nikonov

Date: 28th October 2021 and 11th November 2021

Quantum topology seminar

We show that parities on virtual knots come from invariant 1-cycles on the arcs of knot diagrams.

In turn, the invariant cycles are determined by quasi-indices on the crossings of the diagrams.

The found connection between the parities and the (quasi)-indices allows to define a new series of parities on virtual knots.