arXiv subjects

D. A. Fedoseev

Publications and source records attributed to D. A. Fedoseev.

4 recordsLinked to original sources

arXiv2025

Invariants of Handlebody-Links and Spatial Graphs

A $G-$family of quandles is an algebraic construction which was proposed by A. Ishii, M. Iwakiri, Y. Jang, K. Oshiro in 2013. The axioms of these algebraic systems were motivated by handlebody-knot theory. In the present work we investigate possible constructions which generalise $G-$family of quandles and other similar constructions (for example, $Q-$ and $(G,*,f)-$families of quandles). We provide the necessary conditions under which the resulting object (called an $(X,G,{*_g},f,\otimes,\oplus)-$system) gives a colouring invariant of knotted handlebodies. We also discuss several other modifications of the proposed construction, providing invariants of spatial graphs with an arbitrary (finite) set of values of vertex valency. Besides, we consider several examples which in particular showcase the differences between spatial trivalent graph and handlebody-link theories.

math.GT↗

arXiv2022

On certain properties of pseudo-Riemannian Bertrand manifolds

In the present article we prove and discuss several properties of pseudo-Riemannian Bertrand manifolds, and give an overview of the state of the theory together with problems for future work. In particular, we prove that there are no pseudo-Riemannian completely Bertrand systems -- dynamical systems of movement in a central field on a pseudo-Riemannian 2-dimensional manifold of revolution such that all trajectories of movement are closed.

math.DG↗

arXiv2020

Manifolds of Triangulations, braid groups of manifolds and the groups $Γ_{n}^{k}$

The spaces of triangulations of a given manifold have been widely studied. The celebrated theorem of Pachner~\cite{Pachner} says that any two triangulations of a given manifold can be connected by a sequence of bistellar moves, or Pachner moves, see also~\cite{GKZ,Nabutovsky}. In the present paper we consider groups which naturally appear when considering the set of triangulations with fixed number of simplices of maximal dimension. There are three ways of introducing this groups: the geometrical one, which depends on the metric, the topological one, and the combinatorial one. The second one can be thought of as a ``braid group'' of the manifold and, by definition, is an invariant of the topological type of manifold; in a similar way, one can construct the smooth version. We construct a series of groups $Γ_{n}^{k}$ corresponding to Pachner moves of $(k-2)$-dimensional manifolds and construct a canonical map from the braid group of any $k$-dimensional manifold to $Γ_{n}^{k}$ thus getting topological/smooth invariants of these manifolds.

math.GT↗

arXiv2011

On Long Virtual Biquanles

Virtual quandles with two operations are discussed in the article. Certain knot invariant is constructed and used to distinguish two long virtual knots.

math.GT↗