Place: Chauvenet 201 at 3:45 pm
Time: Wednesday, April 2Speaker: Theodore Stanford, University of Nevada-Reno
Title: Knots Modulo Subgroups of the Pure Braid Group
Abstract: Around 1990, Victor Vassiliev used singularity theory to define a new set of knot invariants. It soon became clear that Vassiliev invariants are closely related to the Jones polynomial and its many generalizations. The collection of all Vassiliev invariants is at least as good at distinguishing knots as the collection of all the Jones-type polynomials. Whether Vassiliev invariants distinguish {\it all\/} knots is still an open question. (It is known that the Jones-type polynomials do not distinguish all knots.)
Using the same formalism as for knots, Vassiliev invariants may be defined (and often computed) for other knot-like things. In the case of braids, it turns out that they are related to the lower central series of the pure braid group $P_k$. ($P_k$ is the kernel of the standard map from the braid group $B_k$ to the symmetric group $S_k$). One may then use purely group-theoretic methods to show that these invariants {\it do\/} distinguish all braids.
If $V$ is the set of all Vassiliev invariants, then $V = \cup_{i=1}^\infty V_i$, where $V_i \subset V_{i+1}$. Each $V_i$ is a finite-dimensional rational vector space, whereas $V$ is infinite-dimensional. An element of $V_i$ which is not an element of $V_{i-1}$ is said to be an invariant of order $i$. Although the space $V$ is at least as strong as the set of knot polynomials, this is not true of any $V_i$, and this may also be shown using the connection with the lower central series of $P_k$.
I will outline the above results, with particular emphasis on the relationship between Vassiliev invariants and $P_k$. I will also try to say a few words about what happens when we consider the derived series (where $\ds_n (G) = [\ds_{n-1} (G), \ds_{n-1} (G)]$) instead of the lower central series (where $\lcs_n (G) = [G, \lcs_{n-1} (G)]$).