James Pommersheim

INTRINSICALLY n-LINKED GRAPHS

For every natural number n, we exhibit a graph with the property that every embedding of it in ℝ3 contains a non-split n-component link. Furthermore, we prove that our graph is minor minimal in the sense that every minor of it has an embedding in ℝ3 that contains no non-split n-component link.

Intrinsically Triple Linked Complete Graphs

Abstract We prove that every embedding of K 10 in R 3 contains a non-split link of three components. We also exhibit an embedding of K 9 with no such link of three components.

Cycles representing the Todd class of a toric variety

In this paper, we describe a way to construct cycles which represent the Todd class of a toric variety. Given a lattice with an inner product we assign a rational number m(s) to each rational polyhedral cone s in the lattice, such that for any toric variety X with fan S, the Todd class of X is the sum over all cones s in S of m(s)[V(s)]. This constitutes an improved answer to an old question of Danilov. In a similar way, beginning with the choice of a complete flag in the lattice, we obtain the cycle Todd classes constructed by Morelli. Our construction is based on an intersection product on cycles of a simplicial toric variety developed by the second-named author. Important properties of the construction are established by showing a connection to the canonical representation of the Todd class of a simplicial toric variety as a product of torus-invariant divisors developed by the first-named author.

Topological symmetry groups of graphs embedded in the 3-sphere

The topological symmetry group of a graph embedded in the 3-sphere is the group consisting of those automorphisms of the graph which are induced by some homeomorphism of the ambient space. We prove strong restrictions on the groups that can occur as the topo- logical symmetry group of some embedded graph. In addition, we characterize the orientation preserving topological symmetry groups of embedded 3-connected graphs in the 3-sphere.

Barvinok's algorithm and the Todd class of a toric variety☆

Abstract In this paper we prove that the Todd class of a simplicial toric variety has a canonical expression as a power series in the torus-invariant divisors. Given a resolution of singularities corresponding to a nonsingular subdivision of the fan, we give an explicit formula for this power series which yields the Todd class. The computational feasibility of this procedure is implied by the additional fact that the above formula is compatible with Barvinok decompositions (virtual subdivisions) of the cones in the fan. In particular, this gives an algorithm for determining the coefficients of the Todd class in polynomial time for fixed dimension. We use this to give a polynomial-time algorithm for computing the number of lattice points in a simple lattice polytope of fixed dimension, a result first achieved by Barvinok.

Values of zeta functions at negative integers, Dedekind sums and toric geometry

In the present paper, we study relations among special values of zeta functions of real quadratic fields, properties of generalized Dedekind sums and Todd classes of toric varieties. The main theme of the paper is the use of toric geometry to explain in a conceptual way properties of the values of zeta functions and Dedekind sums, as well as to provide explicit computations. Both toric varieties and zeta functions associate numerical invariants to cones in lattices, with different motivations and applications. Though we will focus on the case of two-dimensional cones in the present paper, we introduce notation and definitions that are valid for cones of arbitrary dimension. The reason for this added generality is clarity, as well as preparation for the results of a subsequent publication.

Products of Cycles and the Todd Class of a Toric Variety

The purpose of this paper is to show that the Todd class of a simplicial toric variety has a canonical expression in terms of products of torus-invariant divisors. The coefficients in this expression, which are generalizations of the classical Dedekind sum, are shown to satisfy a reciprocity relation which characterizes them uniquely. We achieve these results by giving an explicit formula for the push-forward of a product of cycles under a proper birational map of simplicial toric varieties. Since the introduction of toric varieties in the 1970s, finding formulas for their Todd class has been an interesting and important problem. This is partly due to a well-known application of the Riemann-Roch theorem which allows a formula for the Todd class of a toric variety to be translated directly into a formula for the number of lattice points in a lattice polytope (cf. [Dan]). An example of this application is contained in [Pom], where a formula for the Todd class of a toric variety in terms of Dedekind sums is used to obtain new lattice point formulas. Danilov [Dan] posed the problem of finding a formula for the Todd class of a toric variety in terms of the orbit closures under the torus action. Specifically, he asked if it is possible, given a lattice, to assign a rational number to each cone in the lattice such that given any fan in the lattice, the Todd class of the corresponding toric variety equals the sum of the orbit closures with coefficients given by these assigned rational numbers. Morelli [Mor] showed that such an assignment is indeed possible in a natural way if the coefficients, instead of being rational numbers, are allowed to take values in the field of rational functions on a Grassmannian of linear subspaces of the lattice. However, if it is required that the coefficients be rational numbers invariant under lattice automorphisms, such an assignment is clearly impossible. For example, the nonsingular cone a in 22 generated by (1, 0) and (0, 1) when subdivided by the ray through (1, 1) yields two cones a, and o2 which are both lattice equivalent to a. By additivity, a consequence of the fact that the Todd class pushes forward, we deduce that the coefficient assigned to a must equal 0, which is absurd. In this paper, we show that there is a canonical expression for the Todd class of a simplicial toric variety in terms of products of the torus invariant divisors. Furthermore, this expression is invariant under lattice automorphisms. That is, the coefficient of each product depends only on the set of rays with multiplicities

An Iterated Random Function with Lipschitz Number One

Consider the set of functions

INTRINSICALLY LINKED GRAPHS AND THE SATO–LEVINE INVARIANT

f_{\theta}(x)=|\theta -x|

An Algorithmic Theory of Lattice Points in Polyhedra

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