scholarly journals A Discrete Morse Perspective on Knot Projections and a Generalised Clock Theorem

10.37236/9979 ◽  
2021 ◽  
Vol 28 (3) ◽  
Author(s):  
Daniele Celoria ◽  
Naya Yerolemou
Keyword(s):  
Morse Function ◽  
Finite Sequence ◽  
Graph Laplacian ◽  
Perfect Matchings ◽  
Closed Formula ◽  
Admissibility Condition ◽  
Spanning Forests ◽  
Discrete Morse Function ◽  
Complete Characterisation

We obtain a simple and complete characterisation of which matchings on the Tait graph of a knot diagram induce a discrete Morse function (dMf) on the two sphere, extending a construction due to Cohen. We show these dMfs are in bijection with certain rooted spanning forests in the Tait graph. We use this to count the number of such dMfs with a closed formula involving the graph Laplacian. We then simultaneously generalise Kauffman's Clock Theorem and Kenyon-Propp-Wilson's correspondence in two different directions; we first prove that the image of the correspondence induces a bijection on perfect dMfs, then we show that all perfect matchings, subject to an admissibility condition, are related by a finite sequence of click and clock moves. Finally, we study and compare the matching and discrete Morse complexes associated to the Tait graph, in terms of partial Kauffman states, and provide some computations.

scholarly journals MORSE INEQUALITIES ON CERTAIN INFINITE 2-COMPLEXES

2007 ◽  
Vol 49 (2) ◽  
pp. 155-165 ◽  
Author(s):  
R. AYALA ◽  
L. M. FERNÁNDEZ ◽  
J. A. VILCHES
Keyword(s):  
Finite Number ◽  
Morse Function ◽  
Dimensional Case ◽  
Morse Inequalities ◽  
Discrete Morse Function

AbstractUsing the notion of discrete Morse function introduced by R. Forman for finite cw-complexes, we generalize it to the infinite 2-dimensional case in order to get the corresponding version of the well-known discrete Morse inequalities on a non-compact triangulated 2-manifold without boundary and with finite homology. We also extend them for the more general case of a non-compact triangulated 2-pseudo-manifold with a finite number of critical simplices and finite homology.

scholarly journals Expected sizes of Poisson–Delaunay mosaics and their discrete Morse functions

2017 ◽  
Vol 49 (3) ◽  
pp. 745-767 ◽  
Author(s):  
Herbert Edelsbrunner ◽  
Anton Nikitenko ◽  
Matthias Reitzner
Keyword(s):  
Point Process ◽  
Poisson Point Process ◽  
Probabilistic Models ◽  
Morse Function ◽  
Discrete Point ◽  
Expected Number ◽  
Low Dimensions ◽  
Morse Functions ◽  
Point Set ◽  
Discrete Morse Function

AbstractMapping every simplex in the Delaunay mosaic of a discrete point set to the radius of the smallest empty circumsphere gives a generalized discrete Morse function. Choosing the points from a Poisson point process in ℝn, we study the expected number of simplices in the Delaunay mosaic as well as the expected number of critical simplices and nonsingular intervals in the corresponding generalized discrete gradient. Observing connections with other probabilistic models, we obtain precise expressions for the expected numbers in low dimensions. In particular, we obtain the expected numbers of simplices in the Poisson–Delaunay mosaic in dimensionsn≤ 4.

scholarly journals Min-Max Theory for Cell Complexes

2020 ◽  
Vol 27 (03) ◽  
pp. 447-454
Author(s):  
Lacey Johnson ◽  
Kevin Knudson
Keyword(s):  
Morse Function ◽  
Critical Values ◽  
Discrete Version ◽  
Mountain Pass Lemma ◽  
Smooth Functions ◽  
Alternate Proof ◽  
Cell Complexes ◽  
Lusternik Schnirelmann ◽  
Regular Cell Complex ◽  
Discrete Morse Function

In the study of smooth functions on manifolds, min-max theory provides a mechanism for identifying critical values of a function. We introduce a discretized version of this theory associated to a discrete Morse function on a (regular) cell complex. As applications we prove a discrete version of the mountain pass lemma and give an alternate proof of a discrete Lusternik–Schnirelmann theorem.

scholarly journals Ascending and descending regions of a discrete Morse function

2009 ◽  
Vol 42 (6-7) ◽  
pp. 639-651 ◽  
Author(s):  
Gregor Jerše ◽  
Neža Mramor Kosta
Keyword(s):  
Morse Function ◽  
Discrete Morse Function

scholarly journals Discrete Morse Inequalities on Infinite Graphs

10.37236/127 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Rafael Ayala ◽  
Luis M. Fernández ◽  
José A. Vilches
Keyword(s):  
Morse Function ◽  
Special Kind ◽  
Infinite Graphs ◽  
Morse Inequalities ◽  
Morse Functions ◽  
Critical Elements ◽  
Finite Case ◽  
Discrete Morse Function

The goal of this paper is to extend to infinite graphs the known Morse inequalities for discrete Morse functions proved by R. Forman in the finite case. In order to get this result we shall use a special kind of infinite subgraphs on which a discrete Morse function is monotonous, namely, decreasing rays. In addition, we shall use this result to characterize infinite graphs by the number of critical elements of discrete Morse functions defined on them.

Mathematics of networks

2018 ◽  
Author(s):  
Mark Newman
Keyword(s):  
Random Walks ◽  
Adjacency Matrix ◽  
Graph Partitioning ◽  
Dynamic Networks ◽  
Graph Laplacian ◽  
Network Visualization ◽  
Final Part ◽  
Bipartite Networks ◽  
Component Structure ◽  
Basic Properties

An introduction to the mathematical tools used in the study of networks. Topics discussed include: the adjacency matrix; weighted, directed, acyclic, and bipartite networks; multilayer and dynamic networks; trees; planar networks. Some basic properties of networks are then discussed, including degrees, density and sparsity, paths on networks, component structure, and connectivity and cut sets. The final part of the chapter focuses on the graph Laplacian and its applications to network visualization, graph partitioning, the theory of random walks, and other problems.

scholarly journals Spectral Gap of the Largest Eigenvalue of the Normalized Graph Laplacian

2021 ◽  
Author(s):  
Jürgen Jost ◽  
Raffaella Mulas ◽  
Florentin Münch
Keyword(s):  
Lower Bound ◽  
Bipartite Graph ◽  
Spectral Gap ◽  
Complete Bipartite Graph ◽  
Graph Laplacian ◽  
New Method ◽  
Single Edge ◽  
Largest Eigenvalue ◽  
Complement Graph ◽  
The Largest Eigenvalue

AbstractWe offer a new method for proving that the maxima eigenvalue of the normalized graph Laplacian of a graph with n vertices is at least $$\frac{n+1}{n-1}$$ n + 1 n - 1 provided the graph is not complete and that equality is attained if and only if the complement graph is a single edge or a complete bipartite graph with both parts of size $$\frac{n-1}{2}$$ n - 1 2 . With the same method, we also prove a new lower bound to the largest eigenvalue in terms of the minimum vertex degree, provided this is at most $$\frac{n-1}{2}$$ n - 1 2 .

scholarly journals Rational Krylov methods for fractional diffusion problems on graphs

2021 ◽  
Author(s):  
Michele Benzi ◽  
Igor Simunec
Keyword(s):  
Analytic Function ◽  
Diffusion Equation ◽  
Graph Laplacian ◽  
Fractional Diffusion ◽  
Singular Matrix ◽  
Krylov Methods ◽  
Fractional Powers ◽  
Directed Networks ◽  
Rank One ◽  
Diffusion Problems

AbstractIn this paper we propose a method to compute the solution to the fractional diffusion equation on directed networks, which can be expressed in terms of the graph Laplacian L as a product $$f(L^T) \varvec{b}$$ f ( L T ) b , where f is a non-analytic function involving fractional powers and $$\varvec{b}$$ b is a given vector. The graph Laplacian is a singular matrix, causing Krylov methods for $$f(L^T) \varvec{b}$$ f ( L T ) b to converge more slowly. In order to overcome this difficulty and achieve faster convergence, we use rational Krylov methods applied to a desingularized version of the graph Laplacian, obtained with either a rank-one shift or a projection on a subspace.

scholarly journals Evaluating EYM amplitudes in four dimensions by refined graphic expansion

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
Hongxiang Tian ◽  
Enze Gong ◽  
Chongsi Xie ◽  
Yi-Jian Du
Keyword(s):  
Tree Level ◽  
Tree Amplitude ◽  
Four Dimensions ◽  
Yang Mills ◽  
Negative Helicity ◽  
Spanning Forests ◽  
Single Trace

Abstract The recursive expansion of tree level multitrace Einstein-Yang-Mills (EYM) amplitudes induces a refined graphic expansion, by which any tree-level EYM amplitude can be expressed as a summation over all possible refined graphs. Each graph contributes a unique coefficient as well as a proper combination of color-ordered Yang-Mills (YM) amplitudes. This expansion allows one to evaluate EYM amplitudes through YM amplitudes, the latter have much simpler structures in four dimensions than the former. In this paper, we classify the refined graphs for the expansion of EYM amplitudes into N k MHV sectors. Amplitudes in four dimensions, which involve k + 2 negative-helicity particles, at most get non-vanishing contribution from graphs in N k′ (k′ ≤ k) MHV sectors. By the help of this classification, we evaluate the non-vanishing amplitudes with two negative-helicity particles in four dimensions. We establish a correspondence between the refined graphs for single-trace amplitudes with $$ \left({g}_i^{-},{g}_j^{-}\right) $$ g i − g j − or $$ \left({h}_i^{-},{g}_j^{-}\right) $$ h i − g j − configuration and the spanning forests of the known Hodges determinant form. Inspired by this correspondence, we further propose a symmetric formula of double-trace amplitudes with $$ \left({g}_i^{-},{g}_j^{-}\right) $$ g i − g j − configuration. By analyzing the cancellation between refined graphs in four dimensions, we prove that any other tree amplitude with two negative-helicity particles has to vanish.

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