Georgia Institute of Technology

The Jones slopes of a knot

Mon, 11/30/2009 - 3:05pm, Skiles 269

Stavros Garoufalidis, Georgia Tech, email        Organizer: Stavros Garoufalidis

I will discuss a conjecture that relates the degree of the Jones polynomial of a knot and its parallels with the slopes of incompressible surfaces in the knot complement. I will present examples, as well as computational challenges.

Geometry, computational complexity and algebraic number fields

Mon, 11/23/2009 - 2:00pm, Skiles 269

Hong-Van Le, Mathematical Institute of Academy of Sciences of the Czech Republic        Organizer: Thang Le

In 1979 Valiant gave algebraic analogs to algorithmic complexity problem such as $P \not = NP$. His central conjecture concerns the determinantal complexity of the permanents. In my lecture I shall propose geometric and algebraic methods to attack this problem and other lower bound problems based on the elusive functions approach by Raz. In particular I shall give new algorithms to get lower bounds for determinantal complexity of polynomials over $Q$, $R$ and $C$.

On the Legendrian and transverse classification of cabled knot types

Mon, 11/09/2009 - 2:00pm, Skiles 269

Bulent Tosun, Ga Tech        Organizer: John Etnyre

In 3-dimensional  contact topology one of the main problem  is classifying Legendrian (transverse) knots in certain knot type up to Legendrian ( transverse) isotopy. In particular we want to decide if two (one in case of transverse knots) classical invariants of this knots are complete set of invariants. If it is, then we call this knot type Legendrian (transversely) simple knot type otherwise it is called Legendrian (transversely)  non-simple. In this talk, by tracing the techniques developed by Etnyre and Honda, we will present some  results  concerning  the complete Legendrian and transverse classification of certain cabled knots in the  standard tight contact 3-sphere. Moreover we will provide an infinite family of  Legendrian  and  transversely non-simple prime knots.

Schur Weyl duality and the colored Jones polynomial

Wed, 10/28/2009 - 3:00pm, Skiles 255

Roland van der Veen, University of Amsterdam, email        Organizer: Stavros Garoufalidis

We recall the Schur Weyl duality from representation theory and show how this can be applied to express the colored Jones polynomial of torus knots in an elegant way. We'll then discuss some applications and further extensions of this method.

Triple linking numbers, Hopf invariants and integral formulas for 3-component links

Mon, 10/26/2009 - 2:00pm, Skiles 269

Shea Vela-Vick, Columbia University        Organizer: John Etnyre

To each three-component link in the 3-dimensional sphere we associate a characteristic map from the 3-torus to the 2-sphere, and establish a correspondence between the pairwise and Milnor triple linking numbers of the link and the Pontryagin invariants that classify its characteristic map up to homotopy. This can be viewed as a natural extension of the familiar fact that the linking number of a two-component link is the degree of its associated Gauss map from the 2-torus to the 2-sphere.In the case where the pairwise linking numbers are all zero, I will present an integral formula for the triple linking number analogous to the Gauss integral for the pairwise linking numbers. The integrand in this formula is geometrically natural in the sense that it is invariant under orientation-preserving rigid motions of the 3-sphere.

Exotic smooth structures and knottings in dimension four

Mon, 10/19/2009 - 2:00pm, Skiles 269

Inanc Baykur, Brandeis University        Organizer: John Etnyre

We will introduce new constructions of infinite families of smooth structures on small 4-manifolds and infinite families of smooth knottings of surfaces.

A generalisation of the deformation variety

Mon, 10/12/2009 - 2:05pm, Skiles 269

Henry Segerman, UTexas Austin, email        Organizer: Stavros Garoufalidis

The deformation variety is similar to the representation variety inthat it describes (generally incomplete) hyperbolic structures on3-manifolds with torus boundary components. However, the deformationvariety depends crucially on a triangulation of the manifold: theremay be entire components of the representation variety which can beobtained from the deformation variety with one triangulation but notanother, and it is unclear how to choose a "good" triangulation thatavoids these problems. I will describe the "extended deformationvariety", which deals with many situations that the deformationvariety cannot. In particular, given a manifold which admits someideal triangulation we can construct a triangulation such that we canrecover any irreducible representation (with some trivial exceptions)from the associated extended deformation variety.

Holiday - No Seminar Today

Mon, 10/05/2009 - 2:00pm, -

-, -         Organizer: John Etnyre

Classification of Legendrian twist knots

Mon, 09/28/2009 - 2:00pm, Skiles 269

Vera Vertesi, MSRI        Organizer: John Etnyre

Legendrian knots are knots that can be described only by their projections(without having to separately keep track of the over-under crossinginformation): The third coordinate is given as the slope of theprojections. Every knot can be put in Legendrian position in many ways. Inthis talk we present an ongoing project (with Etnyre and Ng) of thecomplete classification of Legendrian representations of twist knots.

The uniform thickness property and iterated torus knots

Mon, 09/21/2009 - 2:00pm, Skiles 269

Doug LaFountain, SUNY - Buffalo         Organizer: John Etnyre

The uniform thickness property (UTP) is a property of knots embeddedin the 3-sphere with the standard contact structure. The UTP was introduced byEtnyre and Honda, and has been useful in studying the Legendrian and transversalclassification of cabled knot types.  We show that every iterated torus knotwhich contains at least one negative iteration in its cabling sequence satisfiesthe UTP.  We also conjecture a complete UTP classification for iterated torusknots, and fibered knots in general.

Holiday - No Seminar Today

Mon, 09/07/2009 - 2:00pm, -

-, -         Organizer: John Etnyre

Ideal triangulations and algebraic knot invariants

Mon, 08/31/2009 - 2:01pm, Skiles 269

Rinat Kashaev, Section de Mathématiques Université de Genève , email        Organizer: Stavros Garoufalidis

Not yet!

Cube knots and a homology theory from cube diagrams

Mon, 04/20/2009 - 1:00pm, Skiles 269

Scott Baldridge, LSU        Organizer: John Etnyre

In this talk we will introduce the notion of a cube diagram---a surprisingly subtle, extremely powerful new way to represent a knot or link. One of the motivations for creating cube diagrams was to develop a 3-dimensional "Reidemeister's theorem''. Recall that many knot invariants can be easily be proven by showing that they are invariant under the three Reidemeister moves. On the other hand, simple, easy-to-check 3-dimensional moves (like triangle moves) are generally ineffective for defining and proving knot invariants: such moves are simply too flexible and/or uncontrollable to check whether a quantity is a knot invariant or not. Cube diagrams are our attempt to "split the difference" between the flexibility of ambient isotopy of R^3 and specific, controllable moves in a knot projection. The main goal in defining cube diagrams was to develop a data structure that describes an embedding of a knot in R^3 such that (1) every link is represented by a cube diagram, (2) the data structure is rigid enough to easily define invariants, yet (3) a limited number of 5 moves are all that are necessary to transform one cube diagram of a link into any other cube diagram of the same link. As an example of the usefulness of cube diagrams we present a homology theory constructed from cube diagrams and show that it is equivalent to knot Floer homology, one of the most powerful known knot invariants.