Narrow Results

In this paper we prove that two n-gems induce the same manifold if and only if they are linked by a finite sequence of gem moves. A gem move is either a blob move, consisting in the creation or cancellation of an n-dipole, or a clean flip, which is a switch of a pair of edges of the same color that thickens an h-dipole, 1 ≤ h ≤ n - 1, or the inverse operation, which slims an h-dipole, 2 ≤ h ≤ n. Moreover we prove that we can reorder the gem moves, so that all the blob creations precede all clean flips which then precede all the blob cancellations. This reordering is of interest because it is an easy matter to decide whether two gems are linked by a finite sequence of clean flips. As a consequence, if a bound for the number of blob creations is established, then there exists a deterministic finite algorithm to decide whether two gems induce the same manifold or not.

We present a state sum construction of two-dimensional extended Topological Quantum Field Theories (TQFTs), so-called open-closed TQFTs, which generalizes the state sum of Fukuma–Hosono–Kawai from triangulations of conventional two-dimensional cobordisms to those of open-closed cobordisms, i.e. smooth compact oriented 2-manifolds with corners that have a particular global structure. This construction reveals the topological interpretation of the associative algebra on which the state sum is based, as the vector space that the TQFT assigns to the unit interval. Extending the notion of a two-dimensional TQFT from cobordisms to suitable manifolds with corners therefore makes the relationship between the global description of the TQFT in terms of a functor into the category of vector spaces and the local description in terms of a state sum fully transparent. We also illustrate the state sum construction of an open-closed TQFT with a finite set of D-branes using the example of the groupoid algebra of a finite groupoid.

We construct a group ${\mathrm{\Gamma }}_n^4$ corresponding to the motion of points in ${ℝ}^3$ from the point of view of Delaunay triangulations. We study homomorphisms from pure braids on $n$ strands to the product of copies of ${\mathrm{\Gamma }}_n^4$. We will also study the group of pure braids in ${ℝ}^3$, which is described by a fundamental group of the restricted configuration space of ${ℝ}^3$, and define the group homomorphism from the group of pure braids in ${ℝ}^3$ to ${\mathrm{\Gamma }}_n^4$. At the end of this paper, we give some comments about relations between the restricted configuration space of ${ℝ}^3$ and triangulations of the 3-dimensional ball and Pachner moves.

Recently, the first named author defined a 2-parametric family of groups $G_n^k$ [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 $G_n^k$ and dynamical systems led to the discovery of the following fundamental principle: “If dynamical systems describing the motion of $n$ particles possess a nice codimension one property governed by exactly $k$ particles, then these dynamical systems admit a topological invariant valued in $G_n^k$”.

The $G_n^k$ groups have connections to different algebraic structures, Coxeter groups, Kirillov-Fomin algebras, and cluster algebras, to name three. Study of the $G_n^k$ groups led to, in particular, the construction of invariants, valued in free products of cyclic groups. All generators of the $G_n^k$ groups are reflections which make them similar to Coxeter groups and not to braid groups. Nevertheless, there are many ways to enhance $G_n^k$ groups to get rid of this $2$-torsion.

Later the first and the fourth named authors introduced and studied the second family of groups, denoted by ${\mathrm{\Gamma }}_n^k$, 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 ${\mathrm{\Gamma }}_n^k$ naturally appear when considering the set of triangulations with the fixed number of points.

There are two ways of introducing the groups ${\mathrm{\Gamma }}_n^k$: 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 $G_n^k$ and ${\mathrm{\Gamma }}_n^k$ theories and give an overview of recent results in the study of those groups, manifolds, dynamical systems, knot and braid theories.