Download pdf
Morse Theory, Discrete Morse Theory and Applications
- 1 Shanghai Pinghe School, Shanghai, China
* Author to whom correspondence should be addressed.
Abstract
By employing a specific class of smooth functions to study a space, Morse theory establishes deep connections between analysis and topology. It is a classical subject of pure mathematics, originally pioneered by Marston Morse in the 1920s. In this article, we use Morse theory to present a proof of an interesting result on the knots, known as the Fáry-Milnor theorem. We also discuss discrete Morse theory, a subject of applied mathematics developed by Robin Forman in the 1990s, and its application. We focus on elucidating especially the inherent similarity between classical Morse theory and discrete Morse theory
Keywords
Topology, Morse functions, knots, simplicial homology
[1]. István Fáry. Sur lacourbure totaled’une courbe gauche faisant un noeud. Bulletin de la société mathématique de France, 77:128–138, 1949.
[2]. John W Milnor. On the total curvature of knots. Annals of Mathematics, pages 248–257, 1950.
[3]. Anton Petrunin and Stephan Stadler. Six proofs of the fáry-milnor theorem. arXiv preprint arXiv:2203.15137v2, 2023.
[4]. WB Raymond Lickorish. An introduction to knot theory, volume 175. Springer Science & Business Media, 1997.
[5]. Elsa Abbena, Simon Salamon, and Alfred Gray. Modern differential geometry of curves and surfaces with Mathematica. Chapman and Hall/CRC, 2017.
[6]. John W Milnor. Morse theory. Number 51. Princeton university press, 1963.
[7]. Michele Audin, Mihai Damian, and Reinie Erné . Morse theory and Floer homology, volume 2. Springer, 2014.
[8]. John M Lee. Smooth manifolds. Springer, 2012.
[9]. Gerald Teschl. Ordinary differential equations and dynamical systems, volume 140. American Mathematical Soc., 2012.
[10]. Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002.
[11]. Lawrence Craig Evans. Measure theory and fine properties of functions. Routledge, 2018.
[12]. Robin Forman. A user’s guide to discrete morse theory. Séminaire Lotharingien de Combinatoire [electronic only], 48:B48c–35, 2002.
[13]. Arthur B Kahn. Topological sorting of large networks. Communications of the ACM, 5(11):558–562, 1962.
[14]. Nicholas A Scoville. Discrete Morse Theory, volume 90. American Mathematical Soc., 2019.
Cite this article
Dong,Z. (2025). Morse Theory, Discrete Morse Theory and Applications. Theoretical and Natural Science,87,36-51.
Data availability
The datasets used and/or analyzed during the current study will be available from the authors upon reasonable request.
Disclaimer/Publisher's Note
The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of EWA Publishing and/or the editor(s). EWA Publishing and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.
About volume
Volume title: Proceedings of the 4th International Conference on Computing Innovation and Applied Physics
© 2024 by the author(s). Licensee EWA Publishing, Oxford, UK. This article is an open access article distributed under the terms and
conditions of the Creative Commons Attribution (CC BY) license. Authors who
publish this series agree to the following terms:
1. Authors retain copyright and grant the series right of first publication with the work simultaneously licensed under a Creative Commons
Attribution License that allows others to share the work with an acknowledgment of the work's authorship and initial publication in this
series.
2. Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the series's published
version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgment of its initial
publication in this series.
3. Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and
during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See
Open access policy for details).