Rocky Mountain Algebraic Combinatorics Seminar

Random Knot Diagrams

Jason Cantarella
University of Georgia

A knot diagram is an immersion of a single closed loop into the plane, together with over-under information at the crossings. These are interesting combinatorial objects in their own right, and also have the potential to shed some light on old and tricky questions about random knotting. Enumerating knot diagrams is very close to enumerating a subset of planar 4-valent graphs, and closely related problems have been studied by Schaeffer and Zinn-Justin in the context of 2d quantum loop gravity.

In this talk, I'll talk about results from a purely computational enumeration experiment in which we classified all knot diagrams of 10 and fewer crossings using computing resources from the Amazon EC2 service. Then we'll discuss some conjectures which arose from the experiment, and their proofs by my student Harrison Chapman. The computational part is joint work with Harrison Chapman and Matt Mastin, while the proofs are due to Chapman alone.


Simplicial face numbers via extremal graph theory

Michal Adamaszek
University of Copenhagen

One aspect of topological combinatorics is face enumeration, and one of its main problems is to understand how the topology of a spaces affects the face numbers of its simplicial triangulations. For example, Euler proved that any triangulation of S2 satisfies (f0,f1,f2)=(n,3n−6,2n−4), where f0,f1,f2 are the number of vertices, edges and triangles, respectively. Since then various algebraic and topological tools have been developed to study face numbers of spheres and manifolds.

We are interested in this type of question for the natural class of flag complexes, which are just clique complexes of graphs. The good news is that such a complex is completely determined by its 1-skeleton but the bad news is that clique numbers of graphs are not understood nearly as well as face numbers of arbitrary complexes. The structure of sparsest flag triangulations of spheres (a lower bound type of statement) is mostly conjectural and related to the Charney-Davis conjecture and its γ-vector generalizations by Gal. In this talk I will concentrate on the densest flag triangulations (an upper bound statement) of manifolds. I will introduce a method that allows to determine the flag triangulations with maximal (or close to maximal) face numbers and can be tailored to spheres, (homology) manifolds and some classes of pseudomanifolds. It has two ingredients: first, we use tools from extremal graph theory to get a rough structure of the maximizer, and then we rigidify it using whatever topological properties we have at hand.
Joint work with Jan Hladky (Prague).


Weber 201
4–6 pm
Thursday, June 23, 2016
(Refreshments in Weber 117, 3:30–4 pm)
Colorado State University

This is a joint Denver U / UC Boulder / UC Denver / U of Wyoming / CSU seminar that meets biweekly. Anyone interested is welcome to join us at a local restaurant for dinner after the talks.

PDF version

Previous Seminars:

April 29, 2016
Nick Loehr, Jason Williford
April 15, 2016
Alexander Hulpke, Klaus Lux
April 1, 2016
Eamonn O'Brien, Izabella Stuhl
February 19, 2015
James Wilson, Anton Betten
December 4, 2015
Maria Monks Gillespie, Dane Flannery
November 13, 2015
Richard Green, Tim Penttila
October 23, 2015
Christina Boucher, Sylvia Hobart
October 9, 2015
Josh Maglione, Ghodratollah Aalipour
September 25, 2015
Ross McConnell, Henry Adams
September 11, 2015
James B. Wilson, Tim Penttila
May 8, 2015
Amanda Schaeffer Fry, Peter Brooksbank
April 24, 2015
Heide Gluesing-Luerssen, Phil DeOrsey
March 6, 2015
Felice Manganiello, Eric Moorhouse
February 20, 2015
Anton Dzhamay, Anton Betten
February 6, 2015
Alexander Hulpke, Morgan Rodgers
December 5, 2014
Stefaan De Winter, Gretchen Matthews
November 14, 2014
Greg Coxson, Tom Dorsey
October 31, 2014
Octavio Paez Osuna, Sylvia Hobart
October 10, 2014
Takunari Miyazaki, Eric Moorhouse
September 26, 2014
Elissa Ross, Anton Betten
September 12, 2014
Petr Vojtěchovský, Alexander Hulpke
May 9, 2014
Philip DeOrsey, Tim Penttila
April 25, 2014
William J Martin, Jason Williford
April 11, 2014
Victor Pambuccian, George Shakan
March 7, 2014
Nathan Lindzey, Jens Harlander
February 21, 2014
Ross McConnell, Anton Betten
November 22, 2013
Justin Hughes, Josh Maglione

Department of Mathematics
Fort Collins, Colorado 80523