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.
Highlights
Given a graph G embedded in the 2-sphere, denote by G∗ its plane dual, and by Γ(G) the plane graph obtained by overlaying the two graphs
We introduce a set of two moves, called click path and click loop moves; we first prove (Theorem 24) that click path moves induce bijections between the sets of perfect discrete Morse function (dMf) on S2 with different critical points
We introduce two simplicial complexes associated to the set of partial Kauffman states, the matching and discrete Morse complexes
Summary
Given a graph G embedded in the 2-sphere, denote by G∗ its plane dual, and by Γ(G) the plane graph obtained by overlaying the two graphs. Our starting point is to completely characterise the set of possible dMfs arising from a generalised version of Cohen’s construction (Theorem 12) This will allow us (Theorem 16) to prove the existence of a bijection between dMfs arising this way and rooted spanning orthogonal forests in the black and white graphs of the diagram, induced by a generalised version of Kauffman states, called partial Kauffman states. We introduce a set of two moves, called click path and click loop moves; we first prove (Theorem 24) that click path moves induce bijections between the sets of perfect dMfs on S2 with different critical points This implies immediately that the image of the KPW correspondence only consists of perfect dMfs, regardless of the adjacency condition; (Corollary 25) every perfect dMf arises uniquely as the image of precisely one choice of spanning tree and one vertex of each colour. It remains an open question how to extract more information from these complexes associated to Γ(D) and whether this information can be related to interesting features of the knot
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