Stephen Huggett

Polynomial invariants of knots and graphs

I am interested in polynomial invariants of knots and graphs, and Mike Eastwood (Adelaide) and I discovered a representation of the chromatic polynomial of a graph as the Euler characteristic of a corresponding family of manifolds. Some of the key properties of the chromatic polynomials can be derived from simple algebraic topological methods applied to the manifolds. A preprint is available on my articles page, and the paper itself has appeared in the European Journal of Combinatorics.

In a separate piece of work, I have been exploring the relationship between the two "obvious" ways of getting from the Tutte polynomial of a planar graph to the Jones polynomial of the corresponding link. It turns out that hidden within this relationship is the tensor product of the cycle matroids of graphs, which provides a very interesting explanation of the inability of the Jones polynomial to see knot mutations. A preprint is available on my articles page, and the article itself has appeared in the Journal of Knot Theory and its Ramifications.

Iain Moffatt (South Alabama) and I have recently formulated a definition of 2-decompositions of ribbon graphs, which generalises both 2-sums and tensor products of graphs. Furthermore, we give formulae for the Bollobás-Riordan polynomial of such a 2-decomposition, and derive the classical Brylawski formula for the Tutte polynomial of a tensor product as a (very) special case. A preprint is available on my articles page, and the article itself is to appear in the Annals of Combinatorics.

Natalia Virdee and I have recently written a preprint exploring the set of state graphs of a link diagram, showing how the Seifert graphs (arising from orientations of the link diagram) may be constructed directly from the Thistlethwaite (or chess board) graph. We also described in detail the family of Thistlethwaite graphs coming from taking "r-fold parallels" of the original link diagram. The preprint is available on my articles page.

History of geometry

My student Nicole Bloye and I are making a detailed analysis of the ways in which the meaning of geometry has changed from the classical Greek period to the present day. The Euclidean focus on ruler and compass constructions was intriguing, and had profound consequences. We are enjoying exploring these, initially through the works of Bos, Fowler, Hartshorne, Knorr, and Whiteside.

Twistor theory

I find the interplay between the structure of space-time and ideas from algebraic topology and complex differential geometry extremely intriguing, and I am very pleased to have been able to work with Roger Penrose and other members of the mathematical physics research group at the Mathematical Institute, Oxford. I was on the organising committee for the conference, Geometric Issues in the Foundations of Science, held in St John's College Oxford in June 1996, in honour of Roger Penrose. A volume of proceedings, The Geometric Universe, was published by Oxford University Press in 1998: see my books page.

My principal research interest here is in the geometrical and topological structures underlying the twistor description of space-time fields. In particular, my work has concentrated on the research programmes in spin networks, twistor diagrams, and twistor conformal field theory, all of which in various ways probe the twistor description of the interactions between space-time fields.

For example, Franz Müller (Zürich) and I have published an article on sequential twistor diagrams; we were fortunate to have been invited to the Spaces of geodesics and complex structures in general relativity and differential geometry school, Erwin Schrödinger Institute for Mathematical Physics, Vienna, in Spring 1997.

Other branches of twistor geometry also fascinate me, though. Sergei Merkulov (Stockholm) and I published an article on the application of twistor techniques in constructing spaces with exotic holonomies.

In recent years there has been a flurry of activity stimulated by a preprint by Edward Witten (hep-th/0312171) on a string theory in twistor space. Zvi Bern, Philip Candelas, Lionel Mason, Xenia de la Ossa, and I organised a London Mathematical Society workshop at the Mathematical Institute, Oxford in January 2005. Click here for the programme and files of the transparencies of the lectures or here for the notes of my lecture.