Satyan L. Devadoss

Note: A realization of graph associahedra

Given any finite graph, we offer a simple realization of its corresponding graph associahedron polytope using integer coordinates.

Marked tubes and the graph multiplihedron

Given a graph G , we construct a convex polytope whose face poset is based on marked subgraphs of G . Dubbed the graph multiplihedron, we provide a realization using integer coordinates. Not only does this yield a natural generalization of the multiplihedron, but features of this polytope appear in works related to quilted disks, bordered Riemann surfaces and operadic structures. Certain examples of graph multiplihedra are related to Minkowski sums of simplices and cubes and others to the permutohedron. 52B11; 18D50, 55P48

Shape deformation in continuous map generalization

Given a collection of regions on a map, we seek a method of continuously altering the regions as the scale is varied. This is formalized and brought to rigor as well-defined problems in homotopic deformation. We ask the regions to preserve topology, area-ratios, and relative position as they change over time. A solution is presented using differential methods and computational geometric techniques. Most notably, an application of this method is used to provide an algorithm to obtain cartograms.

Pseudograph associahedra

Given a simple graph G, the graph associahedron KG is a simple polytope whose face poset is based on the connected subgraphs of G. This paper defines and constructs graph associahedra in a general context, for pseudographs with loops and multiple edges, which are also allowed to be disconnected. We then consider deformations of pseudograph associahedra as their underlying graphs are altered by edge contractions and edge deletions.

Compatible triangulations and point partitions by series-triangular graphs

We introduce series-triangular graph embeddings and show how to partition point sets with them. This result is then used to prove an upper bound on the number of Steiner points needed to obtain compatible triangulations of point sets. The problem is generalized to finding compatible triangulations for more than two point sets and we show that such triangulations can be constructed with only a linear number of Steiner points added to each point set. Moreover, the compatible triangulations constructed by these methods are regular triangulations.

Convex polytopes from nested posets

Motivated by the graph associahedron K G , a polytope whose face poset is based on connected subgraphs of G , we consider the notion of associativity and tubes on posets. This leads to a new family of simple convex polytopes obtained by iterated truncations. These generalize graph associahedra and nestohedra, even encompassing notions of nestings on CW-complexes. However, these poset associahedra fall in a different category altogether than generalized permutohedra.

Juggling braids and links

Using a simplistic model of juggling based on physics, a natural map is constructed from the set of periodic juggling patterns (or site swaps) to links. We then show that all topological links can be juggled. 1. Juggling sequences The art of juggling has been around for thousands of years. Over the past quarter of a century, the interplay between juggling and mathematics has been well studied. There has even been a book (6) devoted to this relationship, dealing with several combinatorial ideas. Numerous juggling software is also available; in particular, Lipson and Wrights elegant and wonderful JuggleKrazy (4) program helped motivate much of this paper. Most of the information useful to juggling can be accessed via the Juggling Information Service webpage (3). The goal of this paper is to construct and study a map from juggling sequences to topological braids. An early form of this idea providing motivation can be found in the work of Tawney (7), where he looks at some classic juggling patterns. In our discussion, we remove everything that is not mathematically relevant. Thus, as- sume the juggler in question is throwing identical objects, referred to as balls. By convention, there are some basic rules we adhere to in juggling. J1. The balls are thrown to a constant beat, occurring at certain equally-spaced discrete moments in time. J2. At a given beat, at most one ball gets caught and then thrown instantly. J3. The hands do not move while juggling. J4. The pattern in which the balls are thrown is periodic, with no start and no end to this pattern. J5. Throws are made with one hand on odd-numbered beats and the other hand on even-numbered beats. A throw of a ball which takes k beats from being thrown to being caught is called a k-throw. Condition J5 implies that when k is even (or odd), a k-throw is caught with the same (or opposite) hand from which it was thrown. Thus, a 5-throw starting in the left hand would end in the right hand 5 beats later, while a 4-throw starting in the left hand would end back in the left hand after 4 beats. In this notation, a 0-throw is a placeholder so that an empty

Moduli Spaces of Punctured Poincaré Disks

The Tamari lattice and the associahedron provide methods of measuring associativity on a line. The real moduli space of marked curves captures the space of such associativity. We consider a natural generalization by considering the moduli space of marked particles on the Poincare disk, extending Tamari’s notion of associativity based on nesting. A geometric and combinatorial construction of this space is provided, which appears in Kontsevich’s deformation quantization, Voronov’s swiss-cheese operad, and Kajiura and Stasheff’s open-closed string theory.

Skeletal configurations of ribbon trees

The straight skeleton construction creates a straight-line tree from a polygon. Motivated by moduli spaces from algebraic geometry, we consider the inverse problem of constructing a polygon whose straight skeleton is a given tree. We prove there exists only a finite set of planar embeddings of a tree appearing as straight skeletons of convex polygons. The heavy lifting of this result is performed by using an analogous version of Cauchys arm lemma. Computational issues are also considered, uncovering ties to a much older angle bisector problem.