Directions and maps for UCSB and the Mathematics Department.
KITP's guide to local accomodation
Time: Tu 3:304:30 (except as noted)
Place: South Hall 6520 (except as noted)
Date  Speaker  Home Institution  Abstract 
30 Sept  Mike Williams  UCSB  A knot in the 3sphere has tunnel number one if the exterior of this knot admits a genus 2 Heegaard splitting. In the 1980's, John Berge exhibited infinitely many tunnel number one knots that admit lens space Dehn surgeries. These knots are commonly referred to as "Berge knots", and they contain infinitely many hyperbolic knots. It has been conjectured that Berge knots contain all knots in the 3sphere that admit lens space surgeries; this socalled "Berge Conjecture" is still open, even when restricted to the class of tunnel number one knots. I will briefly review Dehn surgery and Heegaard splittings, then discuss workinprogress to address the Berge Conjecture for tunnel number one knots. 
7 Oct.  Dorothy Buck 
The central axis of the famous DNA double helix is often topologically constrained or even circular. The topology of this axis can influence which proteins interact with the underlying DNA. Subsequently, in all cells there are proteins whose primary function is to change the DNA axis topology  for example converting a torus link into an unknot. Additionally, there are several protein families that change the axis topology as a byproduct of their interaction with DNA. This talk will describe typical DNA conformations, and the families of proteins that change these conformations. I'll present a few examples illustrating how Dehn surgery methods have been useful in understanding certain DNAprotein interactions, and discuss the most common topological techniques used to attack these problems. 

14 Oct.  Ken Millett  UCSB  Collectively, Diao, Pippenger, Sumners, and Whittington proved the DelbruckFrischWasserman conjecture that the probability that a selfavoiding random walk or equilateral polygon contains a knot goes to one as the number of edges goes to infinity. A slipknot is defined to be a knotted segment of a walk or polygon that is contained in a larger unknotted segment. Using closures of a polygonal segment to the sphere at infinity, one has a definition of knotting of the segment. We will prove the knotting theorems and to extend them to show that the probability that a selfavoiding random walk or equilateral polygon, in 3space or the simple cubic lattice, contains a slipknot goes to one as the number of edges goes to infinity. 
21 Oct.  Sandra Ritz  USC  We will begin with an overview of the Burau representation of the braid group. This will be followed by an introduction to a contact category on 3manifolds, with a brief discussion of its relation to the braid group. 
28 Oct.  Ilesanmi Adeboye  UCSB  We will discuss progress in developing an explicit lower bound for the volume of a hyperbolic orbifold in dimension 4. 
4 Nov.  Loretta Bartolini  OKSU  Heegaard splittings along orientable surfaces are wellknown in 3manifold theory: the manifold is split into a pair of handlebodies, the embedded discs for which can be used combinatorially to obtain information about both the splitting and the manifold. However, when a nonorientable surface is used in an orientable manifold, the associated Heegaard splitting is onesided and a single handebody is obtained. There are many natural parallels between one and twosided Heegaard splittings, however there are striking and farreaching differences: the presence of singular meridian discs; and, the connection with Z_2 homology. Both properties serve to hamper existing methods, while offering new approaches. Given the direct connection between geometrically incompressible splittings and Z_2 homology classes of the manifold, a finer degree of control of onesided splitting surfaces can be established over their twosided counterparts. In particular, the geometrically incompressible onesided Heegaard splittings of even Dehn fillings of Figure 8 knot space can be explicitly constructed. This involves a result about the behavior of incompressible nonorientable surfaces under Dehn filling, which shows a marked difference from that of either twosided splittings or incompressible surfaces. When considering the global properties of onesided splittings, one is motivated by the twosided precedent, which provides clear fundamental results to pursue. Whilst existence and finite stable equivalence have been known for some time, versions of the key results that progressed the field of twosided splittings have been lacking. One of the first of these  Waldhausen's classification of splittings of S^3  finds an analogue in onesided splittings of RP^3. Whilst there are many alternative proofs for the S^3 result, it is the original that offers a natural generalization to the onesided case. 
11 Nov  Holiday  UCOP  
18 Nov.  Martin Scharlemann  UCSB  Suppose F is a compact orientable surface, K is a knot in F x I, and (F x I)_{surg} is the 3manifold obtained by some nontrivial surgery on K. If F x {0} compresses in (F x I)_{surg}, then there is an annulus in F x I with one end K and the other end an essential simple closed curve in F x{0}. Moreover, the end of the annulus at K determines the surgery slope. An application: suppose M is a compact orientable 3manifold that fibers over the circle. If surgery on a knot K in M yields a reducible manifold, then either

25 Nov.  Claire Levaillant  Caltech  I will study the cases of reducibility of a representation of degree $\frac{n(n1)}{2}$ introduced by Lawrence and Krammer and used by Bigelow to show the linearity of the Braid group on $n$ strands. This representation that is based on two parameters $q$ and $t$ is known to be generically irreducible. We show that when the parameters are specialized to certain complex values, the representation becomes reducible. I will give all the values of the parameters for which the representation is reducible, the dimensions of the invariant subspaces and if time allows their spanning vectors. To do so, I use a knot theoretic approach to construct a representation of the BMW algebra of type $A_{n1}$ in the LawrenceKrammer space. As a representation of the Braid group on $n$ strands, my representation is equivalent to the LawrenceKrammer representation. When the representation is reducible, the action on a proper invariant subspace is an IwahoriHecke algebra action. This allows us to give the complex values of the parameters $l$ and $m$ of the BMW algebra for which my representation is reducible and hence deduce the complex values of the parameters $q$ and $t$ for which the LawrenceKrammer representation is reducible. On the way we show that for these values of the parameters $l$ and $m$ and other values, the BMW algebra is not semisimple. 
2 Dec.  Emille Davie  UCSB 
