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Series: Geometry Topology Seminar

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.

Series: Geometry Topology Seminar

Series: Geometry Topology Seminar

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.

Series: Geometry Topology Seminar

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.

Series: Geometry Topology Seminar

A closed hyperbolic 3-manifold $M$ determines a fundamental classin the algebraic K-group $K_3^{ind}(C)$. There is a regulator map$K_3^{ind}(C)\to C/4\Pi^2Z$, which evaluated on the fundamental classrecovers the volume and Chern-Simons invariant of $M$. The definition of theK-groups are very abstract, and one is interested in more concrete models.The extended Bloch is such a model. It is isomorphic to $K_3^{ind}(C)$ andhas several interesting properties: Elements are easy to produce; thefundamental class of a hyperbolic manifold can be constructed explicitly;the regulator is given explicitly in terms of a polylogarithm.

Series: Geometry Topology Seminar

After reviewing a few techniques from the theory of confoliation in dimension three we will discuss some generalizations and certain obstructions in extending these techniques to higher dimensions. We also will try to discuss a few questions regarding higher dimensional confoliations.

Series: Geometry Topology Seminar

Series: Geometry Topology Seminar

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Series: Geometry Topology Seminar

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.

Series: Geometry Topology Seminar

We will define the sutured version of embedded contact homology for sutured contact 3-manifolds. After that, we will show that the sutured version of embedded contact homology of S^1\times D^2, equipped with 2n sutures of integral or infinite slope on the boundary, coincides with the sutured Floer homology.