Pachner Move 3->3 and Affine Volume-Preserving Geometry in R^3

What discuss the problem of obtaining new manifold invariants via different analogues of 6j-symbols and the torsion of acyclic complexes.

This is the first in a series of papers where we will derive invariants of three-manifolds and framed knots in them from the geometry of a manifold pseudotriangulation put in some way in a four-dimensional Euclidean space. Thus, the elements of the pseudotriangulation acquire Euclidean geometric values such as volumes of different dimensions and various kinds of angles. Then we construct an acyclic complex made of differentials of these geometric values, and its torsion will ...

A cohomology theory for "odd polygon" relations -- algebraic imitations of Pachner moves in dimensions 3, 5, ... -- is constructed. Manifold invariants based on polygon relations and nontrivial polygon cocycles are proposed. Example calculation results are presented.

A pants-block decomposition of a 3-manifold is similar to a triangulation of a 3-manifold in many aspects. In this paper we show that any two pants-block decompositions of a 3-manifold are related by a finite sequence of moves which are called P-moves. The P-moves between pants-block decompositions are similar to the Pachner moves between triangulations. Moreover, we also give a list of types of P-moves. The main tools we used in this paper are the Morse 2-functions, Reeb com...

We demonstrate the triangulability of compact 3-dimensional topological pseudomanifolds and study the properties of such triangulations, including the Hauptvermutung and relations by Alexander star moves and Pachner bistellar moves. We also provide an application to state-sum invariants of 3-dimensional topological pseudomanifolds

Udo Pachner proved that all simplicial manifolds which are homeomorphic can be transformed into each other by a sequence of simple transformations now commonly called "Pachner moves". For a fixed dimension there are only finitely many types of Pachner moves. This makes it possible to identify invariants by proving the invariance only for a finite number of transformations. This fact has proved useful for various applications in p.l. topology and in loop quantum gravity theory...

A common choice for the evolution of the knotted graphs in loop quantum gravity is to use the Pachner moves, adapted to graphs from their dual triangulations. Here, we show that the natural way to consistently use these moves is on framed graphs with edge twists, where the Pachner moves can only be performed when the twists, and the vertices the edges are incident on, meet certain criteria. For other twists, one can introduce an algebraic object, which allow any knotted graph...

A key result in computational 3-manifold topology is that any two triangulations of the same 3-manifold are connected by a finite sequence of bistellar flips, also known as Pachner moves. One limitation of this result is that little is known about the structure of this sequences; knowing more about the structure could help both proofs and algorithms. Motivated by this, we show that there must be a sequence that satisfies a rigid property that we call "semi-monotonicity". We a...

Any two geometric ideal triangulations of a cusped complete hyperbolic $3$-manifold $M$ are related by a sequence of Pachner moves through topological triangulations. We give a bound on the length of this sequence in terms of the total number of tetrahedra and a lower bound on dihedral angles. This leads to a naive but effective algorithm to check if two hyperbolic knots are equivalent, given geometric ideal triangulations of their complements. Given a geometric ideal triangu...

In this paper we present a way to define a set of orthocenters for a triangle in the n-dimensional space R^{n} and we will see some analogies of these orthocenters with the classic orthocenter of a triangle in the Euclidean plane.