On groups Gnk and Γnk: A study of manifolds, dynamics, and invariants
| Autor: | Seongjeong Kim, D. A. Fedoseev, Vassily Olegovich Manturov, Igor Nikonov |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2021 |
| Předmět: |
Pure mathematics
Dynamical systems theory regular triangulation General Mathematics small cancellation regular triangulations braid planarity dynamical system knot kirillov–fomin algebra QA1-939 group Invariant (mathematics) Geometry and topology Mathematics diagram manifold Group (mathematics) Coxeter group coxeter groups Manifold pachner move gnk group Knot theory invariant γnk group manifold of triangulations gale diagram diamond lemma Knot (mathematics) |
| Zdroj: | Bulletin of Mathematical Sciences, Vol 11, Iss 2, Pp 2150004-1-2150004-155 (2021) |
| ISSN: | 1664-3615 1664-3607 |
| DOI: | 10.1142/S1664360721500041 |
| Popis: | Recently, the first named author defined a 2-parametric family of groups [Formula: see text] [V. O. Manturov, Non–reidemeister knot theory and its applications in dynamical systems, geometry and topology, preprint (2015), arXiv:1501.05208]. Those groups may be regarded as analogues of braid groups. Study of the connection between the groups [Formula: see text] and dynamical systems led to the discovery of the following fundamental principle: “If dynamical systems describing the motion of [Formula: see text] particles possess a nice codimension one property governed by exactly [Formula: see text] particles, then these dynamical systems admit a topological invariant valued in [Formula: see text]”. The [Formula: see text] groups have connections to different algebraic structures, Coxeter groups, Kirillov-Fomin algebras, and cluster algebras, to name three. Study of the [Formula: see text] groups led to, in particular, the construction of invariants, valued in free products of cyclic groups. All generators of the [Formula: see text] groups are reflections which make them similar to Coxeter groups and not to braid groups. Nevertheless, there are many ways to enhance [Formula: see text] groups to get rid of this [Formula: see text]-torsion. Later the first and the fourth named authors introduced and studied the second family of groups, denoted by [Formula: see text], which are closely related to triangulations of manifolds. The spaces of triangulations of a given manifolds have been widely studied. The celebrated theorem of Pachner [P.L. homeomorphic manifolds are equivalent by elementary shellings, Europ. J. Combin. 12(2) (1991) 129–145] says that any two triangulations of a given manifold can be connected by a sequence of bistellar moves or Pachner moves. See also [I. M. Gelfand, M. M. Kapranov and A. V. Zelevinsky, Discriminants, Resultants, and Multidimensional Determinants (Birkhäuser, Boston, 1994); A. Nabutovsky, Fundamental group and contractible closed geodesics, Comm. Pure Appl. Math. 49(12) (1996) 1257–1270]; the [Formula: see text] naturally appear when considering the set of triangulations with the fixed number of points. There are two ways of introducing the groups [Formula: see text]: the geometrical one, which depends on the metric, and the topological 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. In this paper, we give a survey of the ideas lying in the foundation of the [Formula: see text] and [Formula: see text] theories and give an overview of recent results in the study of those groups, manifolds, dynamical systems, knot and braid theories. |
| Databáze: | OpenAIRE |
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