arXiv subjects

Lukas Andritsch

Publications and source records attributed to Lukas Andritsch.

4 recordsLinked to original sources

arXiv2020

A note on friezes of type $Λ_p$

A frieze is an array of numbers obeying the unimodular rule. Coxeter showed that a frieze with integer entries corresponds to a triangulation. Recently, Holm and Jørgenson introduced friezes of type $Λ_p$ which correspond to $p$-angulations of a polygon. In this paper we explore the connection between these two types of friezes; in particular we show that the friezes of type $Λ_p$ for $p=4$ and $p=6$ contain integral friezes within them. We also consider the relationships with Farey graphs.

math.CO↗

arXiv2020

The boundary algebra of a GL$_m$-dimer

We consider GL$_m$-dimers of triangulations of regular convex $n$-gons, which give rise to a dimer model with boundary $Q$ and a dimer algebra $Λ_Q$. Let $e_b$ be the sum of the idempotents of all the boundary vertices, and $\mathcal{B}_Q:= e_b Λ_Q e_b$ the associated boundary algebra. In this article we show that given two different triangulations $T_1$ and $T_2$ of the $n$-gon, the boundary algebras are isomorphic, i.e. $e_b Λ_{Q_{T_1}} e_b \cong e_b Λ_{Q_{T_2}} e_b$.

math.RT↗

arXiv2019

Transformed flips in triangulations and matchings

Plane perfect matchings of $2n$ points in convex position are in bijection with triangulations of convex polygons of size $n+2$. Edge flips are a classic operation to perform local changes both structures have in common. In this work, we use the explicit bijection from Aichholzer et al. (2018) to determine the effect of an edge flip on the one side of the bijection to the other side, that is, we show how the two different types of edge flips are related. Moreover, we give an algebraic interpretation of the flip graph of triangulations in terms of elements of the corresponding Temperley-Lieb algebra.

math.CO↗

arXiv2018

Perfect k-colored matchings and (k+2)-gonal tilings

We derive a simple bijection between geometric plane perfect matchings on $2n$ points in convex position and triangulations on $n+2$ points in convex position. We then extend this bijection to monochromatic plane perfect matchings on periodically $k$-colored vertices and $(k+2)$-gonal tilings of convex point sets. These structures are related to a generalization of Temperley-Lieb algebras and our bijections provide explicit one-to-one relations between matchings and tilings. Moreover, for a given element of one class, the corresponding element of the other class can be computed in linear time.

math.CO↗