Click-clock theorem and discrete Morse matchings
Channel: daniele celoria
Date: 2021-12-04

I talk about this paper https://www.combinatorics.org/ojs/index.php/eljc/article/view/v28i3p3  

Comments and questions are very much welcome!

00:00 - Kauffman states
19:50 - Morse matchings and knot projections
38:40 - Counting Morse matchings 
43:55 - Clickclock theorem
1:02:25 - Simplicial complexes

Abstract: 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.

My homepage: https://sites.google.com/view/danieleceloria/