Invited Paper Session Abstracts - Spatial Graph Theory

Thursday, July 27, 1:00 p.m. - 5:00 p.m., Continental Ballroom A

Spatial Graph Theory is a relatively young interdisciplinary field that brings together knot theory, low dimensional topology and geometry, combinatorics, and graph theory, and has applications in chemistry, molecular biology, and biophysics. In addition, because of its combinatorial nature, many problems in Spatial Graph Theory lend themselves well to undergraduate research. For these reasons, faculty at primarily undergraduate institutions as well as those at research universities may be interested in learning about Spatial Graph Theory.

Organizer:
Erica Flapan, Pomona College

Topological Symmetry Groups of Möbius Ladders and the Petersen Graph in \(\mathbb{R}^3\)

1:00 p.m. - 1:20 p.m.
Emille Davie Lawrence, San Francisco University

The study of graphs embedded in \(S^3\) was originally motivated by chemists' need to predict molecular behavior. The symmetries of a molecule can explain many of its chemical properties, however we draw a distinction between rigid and flexible molecules. Flexible molecules may have symmetries that are not merely a combination of rotations and reflections. Such symmetries prompted the concept of the topological symmetry group of a graph embedded in \(S^3\). We will discuss recent work on what groups are realizable as the topological symmetry group for several families of graphs, including the Petersen family and Möbius ladders.

Intrinsic Chirality of Graphs in \(\mathbb{R}^3\) and Other \(3\)-Manifolds

1:30 p.m. - 1:50 p.m.
Hugh Howards, Wake Forest University

We say that a graph \(\Gamma\) embedded in a \(S^3\) is achiral, if there is an orientation reversing homeomorphism \(h\) of \(S^3\) leaving \(\Gamma\) setwise invariant. If no such homeomorphism exists, we say that the embedded graph \(\Gamma\) is chiral. There exist abstract graphs which have the property that all of their embeddings in \(S^3\) are chiral. Such a graph is said to be intrinsically chiral in \(S^3\). This definition can easily be extended to graphs embedded in any 3-manifold, so it is natural to ask whether a graph which is intrinsically chiral in \(S^3\) would necessarily be intrinsically chiral in other 3-manifolds. We survey results about intrinsic chirality of graphs in \(S^3\) and other manifolds.

Alexander Polynomials of Spatial Graphs and Virtual Knots

2:00 p.m. - 2:20 p.m.
Blake Mellor, Loyola Marymount University

The Alexander polynomial is one of the oldest, and most studied, knot invariants. In this talk, we will briefly review the Alexander polynomial and extend it to two generalizations of classical knots: spatial graphs and virtual knots. In spatial graphs, as with knots, the Alexander polynomial is related to \(p\)-colorings of the graph, and can be used to determine whether the graph is \(p\)-colorable. In the realm of virtual knots, we will see how the Alexander polynomial is related to the odd writhe (and, more generally, the writhe polynomial) of a virtual knot.

Realization of Knots and Links in a Spatial Graph

2:30 p.m. - 2:50 p.m.
Kouki Taniyama, Waseda University

A \(\theta_n\) curve graph is a graph with two vertices and n edges joining them. Kinoshita showed the following. For any \(\frac{n(n-1)}{2}\) knots there exists an embedding of a \(\theta_n\) curve graph into space such that the knot types of the \(\frac{n(n-1)}{2}\) embedded cycles coincide with that of the given \(\frac{n(n-1)}{2}\) knots. We will consider the generalization of this result. It is closely related to the theory of Vassiliev invariants and local moves of knots.

Conway-Gordon Type Theorems

3:00 p.m. - 3:20 p.m.
Ryo Nikkuni, Tokyo Woman’s Christian University

Some graphs have the property that no matter how they are embedded in Euclidean space they contain a knot or a link. In 1983, Conway and Gordon proved the following famous theorems: Every spatial complete graph on six vertices contains a two-component link with odd linking number, and every spatial complete graph on seven vertices contains a knot with non-zero Arf invariant. In this talk, we will present an overview of recent developments and ramifications related to the Conway-Gordon theorems in spatial graph theory.

Legendrian Spatial Graphs

3:30 p.m. - 3:50 p.m.
Danielle O’Donnol, Indiana University

This talk will give a brief introduction to contact structures, and focus on Legendrian graphs in the standard contact structure on \(\mathbb{R}^3\). A spatial graph is Legendrian if it is everywhere tangent to the contact structure. We will give an overview of results in this new area of research.

Oriented Matroid Theory and Linear Embeddings of Spatial Graphs

4:00 p.m. - 4:20 p.m.
Elena Pavelescu, University of South Alabama

Matroid theory is an abstract theory of independence introduced by Whitney in 1935. It is a natural generalization of linear independence. Oriented matroids can be thought of as combinatorial abstractions of point configurations over the reals. To every linear (straight-edge) embedding of a graph one can associate an oriented matroid, and the oriented matroid encodes the knotting and linking information in the embedded graph. In this talk, we introduce the basics of oriented matroids and we look at few graph theoretical results which use oriented matroids. In particular, we show that any linear embedding of \(K_9\), the complete graph on nine vertices, contains a non-split link with three components.

Random Linear Embeddings of Spatial Graphs with Applications to Polymers

4:30 p.m. - 4:50 p.m.
Kenji Kozai, Harvey Mudd College

Random knots have been investigated extensively to model knotting behavior of linear polymers like DNA. In general, the complexity of knotting and linking increases as the polymer gets longer. After giving an overview of some of the random knot models that have been studied, we will discuss generalizations into random embeddings of graphs and overview the many questions that arise. As an example, random linear embeddings of a graph can be thought of as a model for the spatial configurations of non-linear molecules and polymers, and one might ask which configurations are typical. Leveraging known and new results about random knotting and linear embeddings of graphs, we show that certain "simple" graphs nearly always show up in their topologically simplest configurations.