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.
We show how to construct homology bases for certain CW complexes in terms of discrete Morse theory and cellular homology. We apply this technique to study certain subcomplexes of the half cube polytope studied in previous works. This involves constructing explicit complete acyclic Morse matchings on the face lattice of…
The paper characterizes discrete Morse functions on knot diagrams and generalizes a clock theorem.
We present a family of complete acyclic Morse matchings on the face lattice of a hypersimplex. Since a hypersimplex is a convex polytope, there is a natural way to form a CW complex from its faces. In a future paper we will utilize these matchings to classify every subcomplex whose reduced homology groups are concentra…
New technique connects graph matching complexes to Morse theory for better topology understanding.
Optimal Morse matchings reveal essential structures of cell complexes which lead to powerful tools to study discrete geometrical objects, in particular discrete 3-manifolds. However, such matchings are known to be NP-hard to compute on 3-manifolds, through a reduction to the erasability problem. Here, we refine the stu…
In the case of smooth manifolds, we use Forman's discrete Morse theory to realize combinatorially any Thom-Smale complex coming from a smooth Morse function by a couple triangulation-discrete Morse function. As an application, we prove that any Euler structure on a smooth oriented closed 3-manifold has a particular rea…
New knots found with Seifert genus not matching minimal genus Seifert surfaces.
1) We introduce random discrete Morse theory as a computational scheme to measure the complicatedness of a triangulation. The idea is to try to quantify the frequence of discrete Morse matchings with a certain number of critical cells. Our measure will depend on the topology of the space, but also on how nicely the spa…
Algorithm simplifies Khovanov homology computations for 4-strand torus links.
Researchers use discrete Morse theory to improve the topology of matching complexes of complete graphs.
We construct a new class of maximal acyclic matchings on the Salvetti complex of a locally finite hyperplane arrangement. Using discrete Morse theory, we then obtain an explicit proof of the minimality of the complement. Our construction provides interesting insights also in the well-studied case of finite arrangements…
We pursue the analogy of a framed flow category with the flow data of a Morse function. In classical Morse theory, Morse functions can sometimes be locally altered and simplified by the Morse moves. These moves include the Whitney trick which removes two oppositely framed flowlines between critical points of adjacent i…
Discrete Morse theory emerged as an essential tool for computational geometry and topology. Its core structures are discrete gradient fields, defined as acyclic matchings on a complex , from which topological and geometrical informations of can be efficiently computed, in particular its homology or Morse-Smale d…
New link invariants from diagram colorings match link widths.
In recent work the author investigates perfect matchings of a bipartite graph obtained from a knot diagram and demonstrates that these correspond to discrete Morse functions on a 2-complex for the 2-sphere. This relationship is expounded below for the opposite audience: those who may be unfamiliar with knots.
Introduces optimization geometrodynamics for dynamic geometric optimization.
We present a complete acyclic matching of the Hasse diagram associated with the face lattice of a hypersimplex. Since a hypersimplex is a convex polytope, there is a natural way to form a CW complex from its faces. We will then utilize this matching along with discrete Morse theory and some topological techniques to cl…
Paper speeds up topological signal identification and cycle matching.
We give an algorithmic computation for the height of Kauffman's clock lattice obtained from a knot diagram with two adjacent regions starred and without crossing information specified. We show that this lattice is more familiarly the graph of perfect matchings of a bipartite graph obtained from the knot diagram by over…
The aim of this paper is to develop a refinement of Forman's discrete Morse theory. To an acyclic partial matching on a finite regular CW complex , Forman introduced a discrete analogue of gradient flows. Although Forman's gradient flow has been proved to be useful in practical computations of homology groups, i…
New method extends discrete Morse theory to simplicial complexes.
Paper constructs continuous families of topological Morse functions.
New proof for discrete Morse theory using combinatorial construction.
Characterizes geodesics on spheres with Morse index bounds and inequalities.
Study continuation maps for Morse fundamental group properties.
For Morse-Smale pairs on a smooth, closed manifold the Morse-Smale-Witten chain complex can be defined. The associated Morse homology is isomorphic to the singular homology of the manifold and yields the classical Morse relations for Morse functions. A similar approach can be used to define homological invariants for i…
The paper introduces Morse theory for Lie groupoids and proves inequalities.
Discrete Morse functions induce shellings with critical tiles corresponding to function's critical faces.
Unified Morse-Bott-Smale chain complex, resolves well-definedness issue.
Random walk constructs Morse functions on surfaces.
The paper develops methods for calculating equivariant homology from Morse functions.
Morse inequalities for noncompact manifolds with group action.
Stability of Yang-Mills connections' Morse indices and nullity in 4D.
We derive a discrete analogue of Morse-Bott theory on CW complexes and use this discrete Morse-Bott function to do some Conley theory analysis. It turns out that our discrete Morse-Bott theory is indeed a generalization of Forman's discrete Morse theory.
New group with non-loxodromic Morse element found.
Morse theory extended to noncompact manifolds with complex geometric data.
Classifies Morse boundaries of 3-manifold groups.
Exponential growth of stable subgroups in Morse geodesics.
The paper studies twisted Morse homology and cohomology on manifolds.
In the present paper, we define Morse-Bott functions on manifolds with boundary which are generalizations of Morse functions and show Morse-Bott inequalities for these manifolds.
In~\cite{rotvandervorst} a homology theory --Morse-Conley-Floer homology-- for isolated invariant sets of arbitrary flows on finite dimensional manifolds is developed. In this paper we investigate functoriality and duality of this homology theory. As a preliminary we investigate functoriality in Morse homology. Functor…
FPP preserves sublinear Morse boundaries in geodesic graphs.
Develops sublinear Morse theory in symmetric spaces.
New pairing defined from Morse complexes for compact manifolds.
We view Dolbeault-Morse-Novikov cohomology H^{p,q}_η(X) as the cohomology of the sheaf Ω_{X,η}^p of η-holomorphic p-forms and give several bimeromorphic invariants. Analogue to Dolbeault cohomology, we establish the Leray-Hirsch theorem and the blow-up formula for Dolbeault-Morse-Novikov cohomology. At last, we conside…
Local-to-global principle for Morse actions on symmetric spaces.
The Morse-Novikov number MN(L) of an oriented link L in the 3-sphere is the minimum number of critical points of a Morse map from the complement of L in the 3-sphere to the circle representing the class of a Seifert surface for L (e.g., the Morse-Novikov number of L is zero if and only if L is fibered). We develop vari…
Relative cup-length defined for non-Morse functions on manifolds.