arXiv subjects

Lowell Abrams

Publications and source records attributed to Lowell Abrams.

6 recordsLinked to original sources

arXiv2019

Which Cubic Graphs have Quadrangulated Spherical Immersions?

We consider spherical quadrangulations -- spherical embeddings of multigraphs, possibly with loops, so that every face has boundary walk of length 4 -- in which all vertices have degree 3 or 4. Interpreting each degree 4 vertex as a crossing, these embeddings can also be thought of as transversal immersions of cubic graphs which we refer to as the {\it extracted graphs}. We also consider quadrangulations of the disk in which interior vertices have degree 3 or 4 and boundary vertices have degree 2 or 3. First, we classify all such quadrangulations of the disk. Then, we provide four methods for constructing spherical quadrangulations, two of which use quadrangulations of the disk as input. Two of these methods provide one-parameter families of quadrangulations, for which we prove that the sequence of isomorphism types of extracted graphs is periodic. We close with a description of computer computations which yielded spherical quadrangulations for all but three cubic multigraphs on eight vertices.

math.CO↗

arXiv2019

New Dualities From Old: generating geometric, Petrie, and Wilson dualities and trialites of ribbon graphs

We develop an algebraic framework for ribbon graphs, revealing symmetry properties of (partial) twisted duality. The original ribbon group action of Ellis-Monaghan and Moffatt restricts self-duality, -petriality, or -triality to the canonical identification of a graph's edges with those of its dual, petrial, or trial, whereas the more natural definition allows any isomorphism. Here we define a new ribbon group action on ribbon graphs, using a semidirect product of the original ribbon group with a permutation group, to take (partial) twists and duals of ribbon graphs while also encoding graph isomorphisms. This brings new algebraic tools to bear on the natural definitions of self-duality etc., as a ribbon graph is a fixed point of this new ribbon group action exactly when it is isomorphic to one of its (partial) twisted duals. With these tools, we prove that every ribbon graph has in its orbit an orientable embedded bouquet, whose (partial) twisted duality properties propagate through the orbit. Thus, (partial) twisted duality properties of all embedded graphs may be analyzed through such bouquets, for which checking isomorphism reduces to checking just dihedral group symmetries. Previous research on self-duality, etc., typically focused on highly symmetric regular maps, but the theory here fully encompasses all cellularly embedded graphs. In contrast to the few, large, very high-genus, self-trial regular maps found by Wilson, and by Jones and Poultin, here we apply our framework to generate all self-trial ribbon graphs on up to seven edges. We also show how a graph's automorphism group may be used to find self-dual, etc., graphs in its orbit, thus exposing the relationship between regularity and the ribbon group action and, answering a question of Jones and Poulton, yielding an infinite family of self-trial graphs not arising as covers or parallel connections of regular maps.

math.CO↗

arXiv2019

A Basic Structure for Grids in Surfaces

A graph $G$ embedded in a surface $S$ is called an $S$-grid when every facial boundary walk has length four, that is, the topological dual graph of $G$ in $S$ is 4-regular. Aside from the case where $S$ is the torus or Klein bottle, an $S$-grid must have vertices of degrees other than four. Let the sequence of degrees other than four in $G$ be called the curvature sequence of $G$. We give a succinct characterization of $S$-grids with nonempty curvature sequence $L$ in terms of graphs that have degree sequence $L$ and are immersed in a certain way in $S$; furthermore, the immersion associated with the $S$-grid $G$ is unique and so our characterization of $S$-grids also partitions the collection of all $S$-grids.

math.CO↗

arXiv1999

Cotensor products of modules

Let C be a coalgebra over a field k and A its dual algebra. The category of C-comodules is equivalent to a category of A-modules. We use this to interpret the cotensor product M \square N of two comodules in terms of the appropriate Hochschild cohomology of the A-bimodule M \otimes N, when A is finite-dimensional, profinite, graded or differential-graded. The main applications are to Galois cohomology, comodules over the Steenrod algebra, and the homology of induced fibrations.

math.RA↗

arXiv1998

Modules, comodules and cotensor products over Frobenius algebras

We characterize noncommutative Frobenius algebras A in terms of the existence of a coproduct which is a map of left A^e-modules. We show that the category of right (left) comodules over A, relative to this coproduct, is isomorphic to the category of right (left) modules. This isomorphism enables a reformulation of the cotensor product of Eilenberg and Moore as a functor of modules rather than comodules. We prove that the cotensor product M \Box N of a right A-module M and a left A-module N is isomorphic to the vector space of homomorphisms from a particular left A^e-module D to N \otimes M, viewed as a left A^e-module. Some properties of D are described. Finally, we show that when A is a symmetric algebra, the cotensor product M \Box N and its derived functors are given by the Hochschild cohomology over A of N \otimes M.

math.RA↗

arXiv1997

The quantum Euler class and the quantum cohomology of the Grassmannians

The Poincare duality of classical cohomology and the extension of this duality to quantum cohomology endows these rings with the structure of a Frobenius algebra. Any such algebra possesses a canonical ``characteristic element;'' in the classical case this is the Euler class, and in the quantum case this is a deformation of the classical Euler class which we call the ``quantum Euler class.'' We prove that the characteristic element of a Frobenius algebra A is a unit if and only if A is semisimple, and then apply this result to the cases of the quantum cohomology of the finite complex Grassmannians, and to the quantum cohomology of hypersurfaces. In addition we show that, in the case of the Grassmannians, the [quantum] Euler class equals, as [quantum] cohomology element and up to sign, the determinant of the Hessian of the [quantum] Landau-Ginzbug potential.

q-alg↗