Forbidden patterns in tropical planar curves and panoptigons

Abstract: Tropical curves in $\mathbb{R}^2$ correspond to metric planar graphs but not all planar graphs arise in this way. We describe several new classes of graphs that cannot occur. For instance, this yields a full combinatorial characterization of the tropically planar graphs of genus at most six. We also define a special family of lattice polytopes namely panoptigons and enumerate all possible panoptigons under mild lattice width constraints and show how they can be used to find a forbidden pattern in tropical planar curves. We also will discuss some possible applications of the classification of panoptigons and ongoing work on suitable generalizations. This talk is based on work in Tewari (2022) and joint work with Michael Joswig (2020) and Ralph Morrison (2021).

algebraic geometrycombinatorics

Audience: researchers in the topic

Online Nottingham algebraic geometry seminar

Series comments: Online geometry seminar, typically held on Thursday. This seminar takes place online via Microsoft Teams on the Nottingham University "Algebraic Geometry" team.

For recordings of past talks, copies of the speaker's slides, or to be added to the Team, please visit the seminar homepage at: kasprzyk.work/seminars/ag.html