The aim of this talk is to show how basic notions traditionally used in the study of "knotted embeddings in dimensions $3$ and $4$", such as covering spaces and representation theory, can have non-trivial applications in combinatorics and statistical mechanics. For example, we will show that for any finite covering $G'$ of a finite edge-weighted graph $G$, the spanning tree partition function on $G$ divides the spanning tree partition function on $G'$ (in the polynomial ring with variables given by the weights). Setting all the weights equal to $1$, this implies a theorem known since 30 years: the number of spanning trees on $G$ divides the number of spanning trees on $G'$. Other examples of such results will be presented.
Joint work (in progress) with Adrien Kassel.[-]

Thurston proved that a post-critically finite branched cover of the plane is either equivalent to a polynomial (that is: conjugate via a mapping class) or it has a topological obstruction. We use topological techniques — adapting tools used tostudy mapping class groups — to produce an algorithm that determines when a branched cover is equivalent to a polynomial, and if it is, determines which polynomial a branched cover is equivalent to.
This is joint work with Jim Belk, Justin Lanier, and Dan Margalit.[-]