Trace Diagrams Research Page

Trace Diagrams

My primary research interest is trace diagrams and their applications to other areas of mathematics. What is a trace diagram? Fundamentally, it's a combinatorial construction that can be identified in a precise way with multilinear functions. If you're a physicist, it's much like a spin network or birdtrack. If you're a knot theorist, it's much like skein algebra, except without the quantization. It's also sometimes called a tensor diagram. The big difference between a trace diagram and these classical constructions is that trace diagrams have edges labeled by matrices, and so the objects of study are almost always functions on matrices.

Applications

I'm a big believer that diagrammatic notation is extremely powerful and has the potential to simplify many classical proofs. Some examples of this capability (for linear algbera) are here. Some other applications include:

• Invariant Theory: Diagrams can be used to describe the functions on matrices which are invariant under simultaneous conjugation (think traces of products of matrices). Trace diagrams are very powerful for demonstrating relations among these functions.
• Geometric Structures: Because invariant theory is vital for understanding character varieties, trace diagrams can be used to describe the moduli space of geometric structures on a surface. (This was the primary direction of my thesis.)

References

Here are the best books on the topic:

• Birdtracks, a webbook by the physicist Predrag Cvitanovic available at http://birdtracks.eu/. This is by far the most thorough reference, but unfortunately is not well-known, especially in the mathematics community. It has some fascinating results for anyone who is interested in the classification of Lie groups. Section 4.9 contains a history of diagrammatic notation which is definitely worth reading.
• Group Theory, by G.E. Stedman (also a physicist). This book is very hard to find and extremely expensive to purchase. Most of the material is also included in Cvitanovic's book.

History

Diagrammatic techniques are notoriously difficult to research from a historical standpoint. The costs of typesetting were extremely high until recently, so many of the early references are buried in PhD theses and hard-to-find references. Roger Penrose invented the term ``spin network'' in the early 1970s for graphs labeled by representations of SU(2) in his work on combinatorial space-time. Today, ``spin network'' is a general term for any graph whose edges represent representations of a particular group and whose nodes are interwiners, or functions, between tensor powers of these representations.

E-Prints and Slides

Work-in-Progress

1. "Computing SL(2,C) Central Functions with Spin Networks" with Sean Lawton. Follow-up to previous article on spin networks and SL(2,C)-character varieties.

Expository

1. "A Not-So-Characteristic Equation: the Art of Linear Algebra" PDF (arXiv: 0712.2058). This article was intended to broaden the audience for diagrammatic techniques, especially in the mathematics community.

Published (or accepted for publication)

1. "Spin networks and SL(2,C)-character varieties" PS/PDF with Sean Lawton (arXiv: math.QA/0511271), to appear in the Handbook of Teichmuller Theory, Vol. II, early 2008. In this paper, we look at the simplest SL(2,C) spin networks (or trace diagrams) labeled by representations and two matrices. These relate to the character varieties of the punctured torus and three-holed sphere, and also to the Fricke-Vogt Theorem. The work depends much on results from Baez ``Spin Networks in Gauge Theory'' and the work of Adam Sikora.

Recent Talks:

1. "Signed Graph Coloring, the Art of Linear Algebra, and a Theorem of Jacobi" PDF Slides, MathFest 2008, 31-Jul-08.
2. "The Character Variety's New Clothes" PDF Slides, AMS Session on Algebra and Number Theory, JMM 2008, 7-Jan-08.
3. "The Art of Linear Algebra" PDF Slides and PDF Handout, USMA Math Department Research Seminar, 5-Dec-07.
4. "Trace Diagrams, Spin Networks, and Spaces of Graphs" PDF Slides and "Diagrammatic Central Functions" PDF Slides, talks at the 7th KAIST Geometric Topology Fair in Gyeongju, Korea, 10-Jul-07.
5. "Trace Diagrams, Surfaces, and Character Varieties" PDF Slides, colloquium at Kansas State University, 15-Apr-07.
6. "Trace Diagrams, Surfaces, and Low-Dimensional Topology" PDF Slides, PhD Defense, 25-Apr-06.