A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
Simplified proof of Lefschetz theorem for PL spheres.
Study PL topology theorems for cubical complexes, solving Habegger and Funar's conjecture.
The paper studies groups related to triangulations and braid groups of manifolds.
3-manifolds have covers with infinitely many ideal triangulations.
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 paper bounds Pachner moves and systoles in hyperbolic 3-manifolds.
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…
Combinatorial description of 3-manifolds using ordered triangulations.
New algebraic rules for 5D shapes based on 3D cocycles.
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.
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.
Machine learning identifies 3-manifold triangulations using isomorphism signatures.
2d dualities linked to 4-simplex triangulation.
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 well-known that the Pachner graph of -vertex triangulated -spheres is connected, i.e., each pair of -vertex triangulated -spheres can be turned into each other by a sequence of edge flips for each . In this article, we study various induced subgraphs of this graph. In particular, we prove tha…
What discuss the problem of obtaining new manifold invariants via different analogues of 6j-symbols and the torsion of acyclic complexes.
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…
The study proves poor ideal three-edge triangulations are minimal for certain 3-manifolds.
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 \& …
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.
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 …
New method finds large counterexamples by selectively exploring triangulations.
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…
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 …
Unimodal sequences of moves connect 3-manifold triangulations.
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…
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…
Study of groups and connects particle dynamics to manifold triangulations.
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…
Method samples triangulations of manifolds using biased random walks.
Using the locally compact abelian group $\BT \times \BZ$, we assign a meromorphic function to each ideal triangulation of a 3-manifold with torus boundary components. The function is invariant under all 2--3 Pachner moves, and thus is a topological invariant of the underlying manifold. If the ideal triangulation has a …
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…
P-moves connect different 3-manifold decompositions.
A classical result by Pachner states that two -dimensional combinatorial manifolds with boundary are PL homeomorphic if and only they can be connected by a sequence of shellings and inverse shellings. We prove that for balanced, i.e., properly -colored, manifolds such a sequence can be chosen such that bala…
In this brief sequel to a previous article, we recall 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 cou…
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 paper studies groups related to braids and triangulations in 3D space.
The Drinfeld double of a finite dimensional Hopf algebra is a quasi-triangular Hopf algebra with the canonical element as the universal -matrix, and one can obtain a ribbon Hopf algebra by adding the ribbon element. The universal quantum invariant of framed links is constructed using a ribbon Hopf algebra. In that c…
New quantum invariant for framed 3-manifolds using ideal triangulations.
This note popularizes a proof for 3-manifold triangulations and spines.
Constructs combinatorial 2D topological field theories from cyclic A-infinity algebras.
Simplified combinatorial descriptions of branched spines for 3-manifolds using primary MP move and sliding moves.
AI generates theorems and proofs for training theorem provers.
Global inverse function theorem proved easily using Riemannian geometry.
A new comparison theorem for geometric spaces.
The paper proves a new theorem in Riemannian geometry and offers a new proof for Toponogov's theorem in Alexandrov geometry.