arXiv subjects
Publications and source records attributed to Satyan L. Devadoss.
We find explicit ridge unfoldings of the three exceptional 4D polytopes (24-cell, 120-cell, 600-cell) that result in overlaps of their facets. These failures bring an end to the full classification of regular polytopes with the all-net property.
We describe new spaces and maps. Our graphical map is a visual and numerical correspondence between spaces of circular electrical networks and circular planar split systems. When restricted to the planar circular electrical case, this graphical map finds the split system uniquely associated with the Kalmanson resistance distance of the dual network, matching the induced split system familiar from phylogenetics. This correspondence is extended to compactifications of the respective spaces, taking cactus networks to the cactus split systems defined herein. The graphical map preserves both network components and cactus structure, allowing an elegant enumeration of induced phylogenetic split systems via combinatorial species. We introduce the global spaces of circular planar electrical networks and circular split systems. These new spaces are also CW complexes, but the 0-cells of each are counted by the Bell numbers as opposed to the Catalan numbers. As species, the two sorts of global cacti are seen to be compositions in complementary ways.
Over a decade ago, it was shown that every edge unfolding of the Platonic solids was without self-overlap, yielding a valid net. We consider this property for regular polytopes in arbitrary dimensions, notably the simplex, cube, and orthoplex. It was recently proven that all unfoldings of the $n$-cube yield nets. We show this is also true for the $n$-simplex and the $4$-orthoplex but demonstrate its surprising failure for any orthoplex of higher dimension.
Given a graph G, the graph associahedron is a simple convex polytope whose face poset is based on the connected subgraphs of G. With the additional assignment of a color palette, we define the colorful graph associahedron, show it to be a collection of simple abstract polytopes, and explore its properties.
We show that every ridge unfolding of an $n$-cube is without self-overlap, yielding a valid net. The results are obtained by developing machinery that translates cube unfolding into combinatorial frameworks. Moreover, the geometry of the bounding boxes of these cube nets are classified using integer partitions, as well as the combinatorics of path unfoldings seen through the lens of chord diagrams.
A classic problem in computational biology is constructing a phylogenetic tree given a set of distances between n species. In most cases, a tree structure is too constraining. We consider a circular split network, a generalization of a tree in which multiple parallel edges signify divergence. A geometric space of such networks is introduced, forming a natural extension of the work by Billera, Holmes, and Vogtmann on tree space. We explore properties of this space, and show a natural embedding of the compactification of the real moduli space of curves within it.
The phylogenetic tree space, introduced by Billera, Holmes, and Vogtmann, is a cone over a simplicial complex. In this short article, we construct this complex from local gluings of classical polytopes, the associahedron and the permutohedron. Its homotopy is also reinterpreted and calculated based on polytope data.
Motivated by the graph associahedron KG, 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.
The orientable cover of the moduli space of real genus zero algebraic curves with marked points is a compact aspherical manifold tiled by associahedra, which resolves the singularities of the space of phylogenetic trees. The resolution maps planar metric trees to their underlying abstract representatives, collapsing and folding an explicit geometric decomposition of the moduli space into cubes. This decomposition endows the resolving space with an interesting canonical pseudometric. The second part of this paper defines a related (stacky) resolution of a space of real quadratic forms, and suggests, perhaps without much justification, that systems of oscillators parametrized by such objects may provide useful models in genomics.
In a previous paper, we showed that the orientable cover of the moduli space of real genus zero algebraic curves with marked points is a compact aspherical manifold tiled by associahedra, which resolves the singularities of the space of phylogenetic trees. In this draft of a sequel, we construct a related (stacky) resolution of a space of real quadratic forms, and suggest, perhaps without much justification, that systems of oscillators parametrized by such objects may may provide useful models in genomics.
Motivated by geometric origami and the straight skeleton construction, we outline a map between spaces of phylogenetic trees and spaces of planar polygons. The limitations of this map is studied through explicit examples, culminating in proving a structural rigidity result.
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 Poincaré 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.
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.
We consider the moduli space of bordered Riemann surfaces with boundary and marked points. Such spaces appear in open-closed string theory, particularly with respect to holomorphic curves with Lagrangian submanifolds. We consider a combinatorial framework to view the compactification of this space based on the pair-of-pants decomposition of the surface, relating it to the well-known phenomenon of bubbling. Our main result classifies all such spaces that can be realized as convex polytopes. A new polytope is introduced based on truncations of cubes, and its combinatorial and algebraic structures are related to generalizations of associahedra and multiplihedra.
The associahedron is a convex polytope whose face poset is based on nonintersecting diagonals of a convex polygon. In this paper, given an arbitrary simple polygon P, we construct a polytopal complex analogous to the associahedron based on convex diagonalizations of P. We describe topological properties of this complex and provide realizations based on secondary polytopes. Moreover, using the visibility graph of P, a deformation space of polygons is created which encapsulates substructures of the associahedron.
There exist natural generalizations of the real moduli space of Riemann spheres based on manipulations of Coxeter complexes. These novel spaces inherit a tiling by the graph-associahedra convex polytopes. We obtain explicit configuration space models for the classical infinite families of finite and affine Weyl groups using particles on lines and circles. A Fulton-MacPherson compactification of these spaces is described and this is used to define the Coxeter operad. A complete classification of the building sets of these complexes is also given, along with a computation of their Euler characteristics.
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 multiphihedron, 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.
Given any finite graph, we offer a simple realization of the graph-associahedron polytope using integer coordinates.