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Arrangements of pseudocircles on surfaces.

Banff International Research Station for Mathematical Innovation and Discovery

A {\em pseudocircle} is an oriented Jordan closed curve on some surface. A finite collection of pseudocircles that pairwise cross in exactly two points is an {\em arrangement of pseudocircles}, and it is {\em strict} if each pseudocircle is a separating curve on the host surface. Following Linhart and Ortner, the combinatorial properties of an arrangement of pseudocircles are encoded in an {\em intersection matrix}, in which each row corresponds to a pseudocircle, and the entries of the row give the cyclic order of its (signed) intersections with the other pseudocircles in the arrangement. Ortner proved that an arrangement of pseudocircles (given as an intersection matrix) can be embedded into the sphere if and only if each of its subarrangements of size four can be embedded in the sphere. In this work we present an extended result, where it was shown that an arrangement of pseudocircles (given as an intersection matrix) is embeddable into the compact orientable surface $S_g$ of genus $g$ if and only if each of its subarrangements of size $4(g+1)$ can be embedded in $S_g$. Joint work with Edgardo Roldan and Gelasio Salazar.

Mathematics; Convex and discrete geometry; Algebraic topology; Combinatorics

video/mp4

Author affiliation: Universidad Autónoma de San Luis Potosí

2017-06-15

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/61975

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/