Stokes posets and serpent nests

We study two different objects attached to an arbitrary quadrangulation of a regular polygon. The first one is a poset, closely related to the Stokes polytopes introduced by Baryshnikov. The second one is a set of some paths configurations inside the quadrangulation, satisfying some specific constraints. These objects provide a generalisation of the existing combinatorics of cluster algebras and nonnesting partitions of type A.Comment: 24 pages, 12 figure

Dynamics of one-dimensional spin models under the line-graph operator

We investigate the application of the line-graph operator to one-dimensional spin models with periodic boundary conditions. The spins (or interactions) in the original spin structure become the interactions (or spins) in the resulting spin structure. We identify conditions which ensure that each new spin structure is stable, that is, its spin configuration minimises its internal energy. Then, making a correspondence between spin configurations and binary sequences, we propose a model of information growth and evolution based on the line-graph operator. Since this operator can generate frustrations in newly formed spin chains, in the proposed model such frustrations are immediately removed. Also, in some cases, the previously frustrated chains are allowed to recombine into new stable chains. As a result, we obtain a population of spin chains whose dynamics is studied using Monte Carlo simulations. Lastly, we discuss potential applications to areas of research such as combinatorics and theoretical biology.Comment: 28 pages, 12 figure

IST Austria Thesis

This thesis considers two examples of reconfiguration problems: flipping edges in edge-labelled triangulations of planar point sets and swapping labelled tokens placed on vertices of a graph. In both cases the studied structures – all the triangulations of a given point set or all token placements on a given graph – can be thought of as vertices of the so-called reconfiguration graph, in which two vertices are adjacent if the corresponding structures differ by a single elementary operation – by a flip of a diagonal in a triangulation or by a swap of tokens on adjacent vertices, respectively. We study the reconfiguration of one instance of a structure into another via (shortest) paths in the reconfiguration graph. For triangulations of point sets in which each edge has a unique label and a flip transfers the label from the removed edge to the new edge, we prove a polynomial-time testable condition, called the Orbit Theorem, that characterizes when two triangulations of the same point set lie in the same connected component of the reconfiguration graph. The condition was first conjectured by Bose, Lubiw, Pathak and Verdonschot. We additionally provide a polynomial time algorithm that computes a reconfiguring flip sequence, if it exists. Our proof of the Orbit Theorem uses topological properties of a certain high-dimensional cell complex that has the usual reconfiguration graph as its 1-skeleton. In the context of token swapping on a tree graph, we make partial progress on the problem of finding shortest reconfiguration sequences. We disprove the so-called Happy Leaf Conjecture and demonstrate the importance of swapping tokens that are already placed at the correct vertices. We also prove that a generalization of the problem to weighted coloured token swapping is NP-hard on trees but solvable in polynomial time on paths and stars

Tilings and Amalgamations

This thesis investigates tilings of the Euclidean Plane and their amalgamations. For any tiling an amalgamation is a tiling produced by joining together tiles of the original tiling. This definition can be extended to cover any suitable adjacency structure, such as a graph. The first chapter of the thesis reviews some of the basic concepts and results in the theory of tilings. Chapter 2 introduces amalgamations, both of tilings and of infinite graphs. Chapter 3 discusses tesseral arithmetic^ and shows how the theory of amalgamations can be used to produce addressing systems of the plane. The second part of the thesis concentrates on classifying and enumerating amalgamators. In chapter 4, we list the possible types of amalgamation of each of the eleven Laves nets. In chapter 5 an algorithm to enumerate a particular class of amalgamations is developed, and the results of running this on a computer are presented. Chapter 6 contains some theoretical results about tiling hierarchies^ sequences of tilings produced by successive amalgamations