Schedule of Topology Seminars: 2008-09

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Time: Tu 3:30-4:30 (except as noted)

Place: South Hall 6520 (except as noted)

Fall quarter 2008

Date Speaker Home Institution
30 Sept Mike Williams UCSB
Lens space surgeries on tunnel number one knots

A knot in the 3-sphere 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 3-sphere that admit lens space surgeries; this so-called "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 work-in-progress to address the Berge Conjecture for tunnel number one knots.

7 Oct. Dorothy Buck
The Topology of DNA-Protein Interactions.

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 by-product 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 DNA-protein interactions, and discuss the most common topological techniques used to attack these problems.

14 Oct. Ken Millett UCSB
Knots and Slipknots in Random Walks and Equilateral Polygons.

Collectively, Diao, Pippenger, Sumners, and Whittington proved the Delbruck-Frisch-Wasserman conjecture that the probability that a self-avoiding 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 self-avoiding random walk or equilateral polygon, in 3-space or the simple cubic lattice, contains a slipknot goes to one as the number of edges goes to infinity.

21 Oct. Sandra Ritz USC
A Categorification of the Burau Representation via Contact Geometry

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 3-manifolds, with a brief discussion of its relation to the braid group.

28 Oct. Ilesanmi Adeboye UCSB
On volumes of hyperbolic 4-orbifolds.

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
One-sided Heegaard splittings of 3-manifolds

Heegaard splittings along orientable surfaces are well-known in 3-manifold 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 non-orientable surface is used in an orientable manifold, the associated Heegaard splitting is one-sided and a single handebody is obtained.

There are many natural parallels between one- and two-sided Heegaard splittings, however there are striking and far-reaching 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 one-sided splitting surfaces can be established over their two-sided counterparts. In particular, the geometrically incompressible one-sided Heegaard splittings of even Dehn fillings of Figure 8 knot space can be explicitly constructed. This involves a result about the behavior of incompressible non-orientable surfaces under Dehn filling, which shows a marked difference from that of either two-sided splittings or incompressible surfaces.

When considering the global properties of one-sided splittings, one is motivated by the two-sided 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 two-sided splittings have been lacking. One of the first of these - Waldhausen's classification of splittings of S^3 - finds an analogue in one-sided 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 one-sided case.

11 Nov Holiday UCOP
18 Nov. Martin Scharlemann UCSB
Surgery on a knot in Surface x I

Suppose F is a compact orientable surface, K is a knot in F x I, and (F x I)surg is the 3-manifold obtained by some non-trivial 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 3-manifold that fibers over the circle. If surgery on a knot K in M yields a reducible manifold, then either

  • the projection K ⊆ M → S1 has non-trivial winding number,
  • K lies in a ball,
  • K lies in a fiber, or
  • K is cabled
25 Nov. Claire Levaillant Caltech
Reducibility of the Lawrence-Krammer representation

I will study the cases of reducibility of a representation of degree $\frac{n(n-1)}{2}$ 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 $A_{n-1}$ in the Lawrence-Krammer space. As a representation of the Braid group on $n$ strands, my representation is equivalent to the Lawrence-Krammer representation. When the representation is reducible, the action on a proper invariant subspace is an Iwahori-Hecke 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 Lawrence-Krammer 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

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