Pachner move 3 ->3 deals with triangulations of four-dimensional manifolds. We present an algebraic relation corresponding in a natural way to this move and based, a bit paradoxically, on three-dimensional geometry.
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48 results for “Pachner Moves”
Quadrangulation movesmath.GT
Transforming quadrangulations on surfaces with moves.
problem Transforming quadrangulations of the same surface.
method Two-dimensional local cubical Pachner moves and non-local subdivision transformations.
result Any two quadrangulations of the same surface can be transformed into one another.
The paper bounds Pachner moves and systoles in hyperbolic 3-manifolds.
problem Bounding Pachner moves and systoles in cusped hyperbolic 3-manifolds.
method Using geometric ideal triangulations and dihedral angles, the paper gives bounds on Pachner moves and systoles.
result Lower bounds on systole length and Pachner move sequence length.
Combinatorial description of 3-manifolds using ordered triangulations.
problem Understanding closed 3-manifolds through ideal triangulations.
method Combining ordered ideal triangulations and Pachner moves.
result Closed 3-manifolds can be described via ordered triangulations and moves.
New algebraic rules for 5D shapes based on 3D cocycles.
problem Creating rules for 5D shapes.
method Using simplicial 3-cocycles to parameterize heptagon relations.
result Parameterized heptagon relations for 5D shapes.
In this paper we describe a procedure to simplify any given triangulation of the 3-sphere using Pachner moves. We obtain an explicit exponential-type bound on the number of Pachner moves needed for this process. This leads to a new recognition algorithm for the 3-sphere.
Algorithm checks if geometrically triangulated manifolds are isometric.
problem Determining if two geometric triangulations of manifolds are isometric.
method Sequence of Pachner moves and barycentric subdivisions with bounds on lengths.
result Bounding the length of transformations between triangulations.
The paper studies groups related to triangulations and braid groups of manifolds.
problem Understanding the structure of triangulations and their invariants.
method Introducing and studying groups and their connections to braid groups.
result Constructs a canonical map from braid groups to , providing topological/smooth invariants.
It is important to have effective methods for simplifying 3-manifold triangulations without losing any topological information. In theory this is difficult: we might need to make a triangulation super-exponentially more complex before we can make it smaller than its original size. Here we present experimental work that…
2d dualities linked to 4-simplex triangulation.
problem Exploring dualities between 2d and 3d theories via 4-manifold geometry.
method Using Pachner moves and supersymmetric half-indices.
result Identified IR dualities and abelian dualities.
Study PL topology theorems for cubical complexes, solving Habegger and Funar's conjecture.
problem Characterize PL homeomorphic cubulations equivalence by Pachner moves.
method Show equivalence to the existence of cobordisms between generic immersions of hypersurfaces.
result Solve Habegger and Funar's conjecture about PL homeomorphic cubulations equivalence.
The Sample Compression Conjecture of Littlestone & Warmuth has remained unsolved for over two decades. This paper presents a systematic geometric investigation of the compression of finite maximum concept classes. Simple arrangements of hyperplanes in Hyperbolic space, and Piecewise-Linear hyperplane arrangements, are …
We extend results of Pachner and Casali to give finite sets of moves relating triangulations of PL manifolds respecting filtrations by locally flat manifolds and stratifications in which a finite family of simple local models exists for neighborhoods of strata.
Unimodal sequences of moves connect 3-manifold triangulations.
problem Understanding the structure of sequences of bistellar flips.
method Examined unimodal sequences of moves that increase and decrease triangulation size.
result Proved that any two one-vertex triangulations are connected by a unimodal sequence of moves.
3-manifolds have covers with infinitely many ideal triangulations.
problem Proving the existence of infinitely many geometric ideal triangulations in certain 3-manifolds.
method Using separability of peripheral subgroups and conjugacy separability theorems.
result Every cusped hyperbolic 3-manifold has a cover with infinitely many geometric ideal triangulations.
What discuss the problem of obtaining new manifold invariants via different analogues of 6j-symbols and the torsion of acyclic complexes.
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 pse…
Matveev and Piergallini independently showed that, with a small number of known exceptions, any triangulation of a three-manifold can be transformed into any other triangulation of the same three-manifold with the same number of vertices, via a sequence of 2-3 and 3-2 moves. We can interpret this as showing that the Pa…
Here are versions of the proofs of two classic theorems of combinatorial topology. The first is the result that piecewise linearly homeomorphic simplicial complexes are related by stellar moves. This is used in the proof, modelled on that of Pachner, of the second theorem. This states that moves from only a finite coll…
The 3D Index is extended to meromorphic functions on triangulated 3-manifolds.
problem Extending the 3D Index to a broader class of triangulated 3-manifolds.
method Assigning a meromorphic function to each ideal triangulation, invariant under Pachner moves, and expanding it into a Laurent series.
result The meromorphic function can be computed from gluing equations and coincides with the 3D Index for ideal triangulations with strict angle structures.
Balanced shellings preserve balancedness in manifold transformations.
problem Preserving balancedness in shellings and inverse shellings of balanced manifolds.
method Established cross-flips and shellings to connect balanced manifolds, preserving balancedness.
result A sequence of cross-flips can connect any two balanced PL homeomorphic manifolds.
The study proves poor ideal three-edge triangulations are minimal for certain 3-manifolds.
problem Finding minimal ideal triangulations for specific 3-manifolds.
method Analyzing properties of poor ideal three-edge triangulations and applying them to construct minimal triangulations.
result Poor ideal three-edge triangulations are proven to be minimal for certain 3-manifolds.
A general state-sum construction of 2-dimensional topological quantum field theories with defectsmath.QA
We derive the general state sum construction for 2D topological quantum field theories (TQFTs) with source defects on oriented curves, extending the state-sum construction from special symmetric Frobenius algebra for 2-D TQFTs without defects (cf. Lauda \& Pfeiffer \cite{LP}). From the extended Pachner moves (Crane \& …
P-moves connect different 3-manifold decompositions.
problem Connecting different 3-manifold decompositions.
method Using P-complex and Morse 2-functions, we show P-moves between pants-block decompositions.
result Any two pants-block decompositions are related by a finite sequence of P-moves.
We introduce a notion of cross-flips: local moves that transform a balanced (i.e., properly -colored) triangulation of a combinatorial -manifold into another balanced triangulation. These moves form a natural analog of bistellar flips (also known as Pachner moves). Specifically, we establish the following the…
It is not completely unreasonable to expect that a computable function bounding the number of Pachner moves needed to change any triangulation of a given 3-manifold into any other triangulation of the same 3-manifold exists. In this paper we describe a procedure yielding an explicit formula for such a function if the 3…
The aim of this paper (inspired from a problem of Habegger) is to describe the set of cubical decompositions of compact manifolds mod out by a set of combinatorial moves analogous to the bistellar moves considered by Pachner, which we call bubble moves. One constructs a surjection from this set onto the the bordism gro…
In this paper we extend the classical theory of combinatorial manifolds to the non-homogeneous setting. NH-manifolds are polyhedra which are locally like Euclidean spaces of varying dimensions. We show that many of the properties of classical manifolds remain valid in this wider context. NH-manifolds appear naturally w…
It was recently shown that there exists an explicit bound for the number of Pachner moves needed to connect any two triangulation of any Haken 3-manifold which contains no fibred sub-manifolds as strongly simple pieces of its JSJ-decomposition. In this paper we prove a generalisation of that result to all knot compleme…
It is not known whether there exists a computable function bounding the number of Pachner moves needed to connect any two triangulation of a compact 3-manifold. In this paper we find an explicit bound of this kind for all Haken 3-manifolds which contain no fibred submanifolds as strongly simple pieces of their JSJ-deco…
Simplified combinatorial descriptions of branched spines for 3-manifolds using primary MP move and sliding moves.
problem Combinatorial descriptions of branched spines for 3-manifolds and their equivalence relations.
method Demonstrated that 16 MP moves on branched spines are derived from a primary MP move, pure sliding moves, and their inverses.
result Simpler combinatorial descriptions for closed 3-manifolds and combed 3-manifolds.
We introduce the notion of a cut cellular surface (CCS), being a surface with boundary, which is cut in a specified way to be represented in the plane, and is composed of 0-, 1- and 2-cells. We obtain invariants of CCS's under Pachner-like moves on the cellular structure, by counting colourings of the 1-cells with elem…
The Pachner graph of 2-spheres is studied, focusing on subgraphs of flag and stacked 2-spheres.
problem Characterize subgraphs of the Pachner graph of 2-spheres.
method Analyzes various induced subgraphs of the Pachner graph of -vertex triangulated 2-spheres, proving connectivity and providing bounds on the number of connected components.
result The subgraph of -vertex flag 2-spheres is connected, while the subgraph of -vertex stacked 2-spheres has at least as many connected components as trees with specific properties.
Paper extends quantum invariant to colored ideal triangulations.
problem Quantum invariants for colored ideal triangulations.
method Uses Heisenberg double and pentagon relation to extend invariant.
result Invariance of colored ideal triangulations under moves.
Machine learning identifies 3-manifold triangulations using isomorphism signatures.
problem Differentiating and classifying 3-manifolds and their Dehn surgeries.
method Training machine learning models on isomorphism signatures derived from 3-manifold triangulations and Pachner graphs.
result Gradient saliency analysis reveals key parts of the language-like encoding scheme.
Method samples triangulations of manifolds using biased random walks.
problem Efficiently sample triangulations of manifolds.
method Biased random walk through Pachner graph with Metropolis-Hastings accept/reject probabilities.
result Samples triangulations at random from chosen probability, estimating rare triangulations.
Simplified proof of Lefschetz theorem for PL spheres.
problem Proving the Lefschetz theorem for PL spheres.
method Using Pachner's Theorem to replace geometric constructions.
result Simpler proof and implications for g-conjecture and Kalai-Sarkaria conjecture.
Study of groups and connects particle dynamics to manifold triangulations.
problem Understanding the relationship between particle dynamics and manifold triangulations.
method Introducing and studying groups and to connect dynamical systems to topological invariants.
result Found a topological invariant valued in for dynamical systems.
This note popularizes a proof for 3-manifold triangulations and spines.
problem Connecting triangulations of 3-manifolds with specific moves.
method Dual viewpoints of triangulations and spines; combinatorial proof replacing general position arguments.
result Unified proof for both closed and non-compact 3-manifolds.
The paper studies groups related to braids and triangulations in 3D space.
problem Understanding the relationship between braids and groups in three-dimensional space.
method Constructs groups and studies homomorphisms from braids to these groups.
result Defines and analyzes groups and their connections to pure braids in 3D space.
Invariants for surfaces with 0-, 1-, and 2-cells are derived from finite 2-groups.
problem Counting colorings of 1- and 2-cells in a surface to define invariants.
method Counting colorings of 1- and 2-cells with elements of a finite 2-group, subject to a fake flatness condition.
result Invariants of cut cellular surfaces extend Yetter's invariants to this class of surfaces.
New method finds large counterexamples by selectively exploring triangulations.
problem Finding small counterexamples in -manifold triangulations.
method Selective enumeration of triangulations using heuristics.
result Found counterexamples to three conjectures about vertex triangulations.
New quantum invariant for framed 3-manifolds using ideal triangulations.
problem Quantum invariants of framed 3-manifolds with vanishing first Betti number.
method Based on ideal triangulations and Hopf algebras, using the pentagon equation and graphical representations.
result Construction of a new quantum invariant for closed framed 3-manifolds.
It is important to have fast and effective methods for simplifying 3-manifold triangulations without losing any topological information. In theory this is difficult: we might need to make a triangulation super-exponentially more complex before we can make it smaller than its original size. Here we present experimental …
A new knot move preserves pass-move equivalence and differs in count.
problem Defining and analyzing a new knot move.
method Introducing the --move and comparing it to pass-move and -move.
result Equivalence under --move for knots is equivalent to pass-move equivalence.
It is well known that any two diagrams representing the same oriented link are related by a finite sequence of Reidemeister moves O1, O2 and O3. Depending on orientations of fragments involved in the moves, one may distinguish 4 different versions of each of the O1 and O2 moves, and 8 versions of the O3 move. We introd…
In this paper, we introduce an equivalence relation on the set of local moves and classify local moves, called the extended -moves, up to the equivalence. Moreover, by inducing a binary relation on the set of equivalence classes of local moves, we show that an extended -move realizes the crossing change or the …
Minimal sets of moves for isotopic knots and trivalent graphs identified.
problem Identifying minimal sets of moves for isotopic knots and trivalent graphs.
method Provided and proved the existence of minimal generating sets of oriented Reidemeister moves for isotopic knots and spatial trivalent graphs.
result Twelve minimal generating sets of oriented Reidemeister moves for isotopic knots and ten for spatial trivalent graphs identified.