Program 2019

Abstract: Twin groups are right angled Coxeter groups that can be thought of as planar analogues of braid groups. Pure twin groups can be defined similar to pure braid groups. We present some recent algebraic and topological observations about these groups.

Abstract: In my talk I will introduce operation cabling on braids. This operation gives a possibility to define a new generating sets of the pure braid groups (classical and virtual). Also one can defines simplicial structures on these groups. In the case of classical pure braid groups F. Cohen and J. Wu proved that the simplicial group $AP_*$ contains Milnor’s simplicial sphere $F[S^1]$ that gives, in particular, interpretation generators homotopy groups of 2-sphere in terms of the pure braids.

Abstract: Gordian complex of knots were introduced by Hirasawa and Uchida. Later this concept was extended to virtual knots. In both cases, Gordian complex was defined using classical unknotting operations. In this talk, we will discuss various outcomes if we consider different unknotting operations in defining the Gordian distance of knots/virtual knots.

Abstract: In this talk we discuss behavior of F-polynomials under connected sums o virtual knots. We show how to construct an infinite family of different connected sums of the same pair of oriented virtual knots, when one of them has \nabla J_n \neq 0.

Abstract: In three-dimensional hyperbolic space, we consider two types of right-angled polyhedra: compact — all vertices are finite, ideal — with all vertices on the absolute. In this talk, we will discuss boundaries of the volume of these polyhedra in terms of the number of vertices and the sizes of the faces, hypotheses about polyhedra with maximum and minimum volume, correlations between the hyperbolic volume of combinatorial invariants of polyhedra.

Abstract: We consider a compact hyperbolic tetrahedron T of general type. It is well known that T is uniquely defined up to isometry either by the set of its dihedral angles or the set of its edge lengths. A Gram matrix G(T) of tetrahedron T consists of cosines of its dihedral angles, while an edge matrix E(T) is formed by hyperbolic cosines of its edge lengths. There are some relations known for a Gram matrix of a hyperbolic tetrahedron T. Also, A. Mednykh and M. Pashkevich (2006) found a formula expressing edge lengths of T in terms of its Gram matrix cofactors. In the present work, we obtain analogous relations but in terms of edge matrix of a compact hyperbolic tetrahedron T.