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.
@article{10_37236_9979,
author = {Daniele Celoria and Naya Yerolemou},
title = {A discrete {Morse} perspective on knot projections and a generalised clock theorem},
journal = {The electronic journal of combinatorics},
year = {2021},
volume = {28},
number = {3},
doi = {10.37236/9979},
zbl = {1479.57004},
url = {http://geodesic.mathdoc.fr/articles/10.37236/9979/}
}
TY - JOUR
AU - Daniele Celoria
AU - Naya Yerolemou
TI - A discrete Morse perspective on knot projections and a generalised clock theorem
JO - The electronic journal of combinatorics
PY - 2021
VL - 28
IS - 3
UR - http://geodesic.mathdoc.fr/articles/10.37236/9979/
DO - 10.37236/9979
ID - 10_37236_9979
ER -
%0 Journal Article
%A Daniele Celoria
%A Naya Yerolemou
%T A discrete Morse perspective on knot projections and a generalised clock theorem
%J The electronic journal of combinatorics
%D 2021
%V 28
%N 3
%U http://geodesic.mathdoc.fr/articles/10.37236/9979/
%R 10.37236/9979
%F 10_37236_9979
Daniele Celoria; Naya Yerolemou. A discrete Morse perspective on knot projections and a generalised clock theorem. The electronic journal of combinatorics, Tome 28 (2021) no. 3. doi: 10.37236/9979
