Tropically planar graphs: counting and constraints

Talk by Ralph Morrison

Tropical plane curves are a combinatorial analog of traditional algebraic plane curves.  Any smooth tropical plane curve contains a distinguished graph called its skeleton; and any graph that is the skeleton of some smooth tropical plane curve is called "tropically planar".  We can quickly show that tropically planar graphs are trivalent, connected, and planar; what more can we say?  In this talk we present the strongest known results on properties of tropically planar graphs, and also present upper and lower bounds on the number of them with fixed Betti number.  This includes joint work with Ayush K. Tewari, and with participants in the 2017 and 2020 iterations of the SMALL REU at Williams College.