  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 $S^2$,
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.
