3-Manifold Seminar: Invariants of spatial graphs

Seminar | September 19 | 2:10-3:30 p.m. | 740 Evans Hall

 Kyle Miller, UC Berkeley

 Department of Mathematics

Knot theory can be generalized to spatial graphs: embeddings of framed topological graphs in $S^3$ up to isotopy. Certain categories support graphical notations for homomorphisms (for instance, elements of $S_n$ as $n$ immersed arcs in the plane), and by reinterpreting a projection of a spatial graph graphically, we may obtain an algebraic invariant.

We will discuss Penrose's graphical tensor notation, the Penrose $\mathfrak {so}(n)$ and $\mathfrak {sl}(n)$ polynomials for abstract cubic graphs, the Jones and Yamada polynomials via the Temperley-Lieb algebra, the Kauffman and Jaeger polynomials via the Birman-Murakami-Wenzl algebra, and the HOMFLY and Murakami polynomials via a Iwahori-Hecke algebra. A running theme will be relationships and coincidences among these polynomials, as well as applications to topics such as the $4$-color theorem and combinatorial embeddings of abstract graphs.