### Seminars

### on knot theory and related topics

The groups Gn k and Gn k and their applications in topology, algebra and geometry

Speaker : V.O. Manturov

Date: 3rd September 2022

Topological Methods in Mathematical Physics

In 2015, the author defined the two-parametric family of groups Gn k , that depend on two integer parameters n > k, and formulated the following principle: “If a dynamical system describing the motion of n particles possesses a nice co-dimension 1 property governed by exactly k particles, then this dynamical system has a topological invariant valued in Gn k .” The first examples describe motions of points on the plane with properties that “3 points are collinear” and “4 points belong to the same circle or line”, which lead to homomorphisms from the pure braid group given by PBn → Gn 3 , and PBn –> Gn k , respectively. In a similar manner we defined other series of groups Gn 4 , that correspond to the configuration space of triangulations.We shall describe various examples and applications of the groups Gn k and Gn 4 in algebra, geometry and topology and formulate some open problems. This is joint work with several collaborators including I.M. Nikonov, S. Kim, D.A. Fedoseev, Z. Wan, J. Wu and H. Yan.

[1] Manturov, V.O., Fedoseev, D.A., Nikonov, I.M. and Kim, S. 2020 Invariants and Pictures: Low-Dimensional Topology and Combinatorial Group Theory. World Scientific.

Title: Analogues of braid groups and Coxeter groups and their applications in topology, algebra and geometry

Speaker: V. O. Manturov

Date: 28 March 2022

Organization: DEFORMATION THEORY SEMINAR

Abstract: A family of groups $G_{n}^{k}$ may be regarded as analogues of braid groups and Coxeter groups. Defining relations for look like group-like versions of the ``tetrahedron relations'' (higher Yang-Baxter-Relations).If dynamical systems describing the motion of $n$ particles possess a nice codimension 1 property governed by exactly $k$ particles, then these dynamical systems admit a topological invariant valued in $G_{n}^{k}$.}These groups have connections to different algebraic structures; in particular, the construction of invariants, valued in free products of cyclic groups. All generators of the groups are reflections but there are many ways to enhance them to get rid of $2$-torsion. Later joint work with I.M.Nikonov introduced and studied a second family of groups, which are closely related to sets of triangulations of manifolds fixed number of points.There are two ways of introducing these groups: geometrical (depending on the metric) and topological. The second one can be thought of as a ``braid group'' of the manifold and is an invariant of the topological type of the manifold; in a similar way, one can construct the smooth version. The talk is a survey giving an overview of recent results related to manifolds, dynamical systems, knots and braids.Many problems and research projects will be formulated at the end of the talk.V.O. Manturov, D.A. Fedoseev, S. Kim, I.M. Nikonov, “On groups Gkn and Γkn: A study of manifolds, dynamics, and invariants”, Bulletin of Mathematical Sciences, 11:02 (2021), 2150004{Book} V.O.Manturov, D.A.Fedoseev, S.Kim, and I.M.Nikonov,Invariants and Pictures:Low-Dimensional Topology and Combinatorial Group Theory,

Record of the talk (Password for access to record : !Rfx&x%1 )

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.

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.

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

# 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

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.

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.