- You are here:
- GT Home
- Home
- News & Events

Series: Geometry Topology Seminar

In this talk we will exhibit many new phenomena in the structure of Legendrian and transverse knots by giving a complete classification of all cables of the positive torus knots. We will also provide two structural theorems to ensure when cable of a Legendrian simple knot type is also Legendrian simple. Part of the results are joint work with John Etnyre and Douglas LaFountain

Series: Geometry Topology Seminar

For knots the hyperbolic geometry of the complement is known to be relatedto itsJones polynomial in various ways. We propose to study this relationship morecloselyby extending the Jones polynomial to graphs. For a planar graph we will showhow itsJones polynomial then gives rise to the hyperbolic volume of the polyhedronwhose1-skeleton is the graph. Joint with Francois Gueritaud and FrancoisCostantino.

Series: Geometry Topology Seminar

The braid group embeds in the mapping class group, and so the symplectic representation of the mapping class group gives rise to a symplectic represenation of the braid group. The basic question Tara Brendle and I are trying to answer is: how can we describe the kernel? Hain and Morifuji have conjectured that the kernel is generated by Dehn twists. I will present some progress/evidence towards this conjecture.

Series: Geometry Topology Seminar

Gromov defined the distortion of an embedding of S^1 into R^3 and asked whether every knot could be embedded with distortion less than 100. There are (many) wild embeddings of S^1 into R^3 with finite distortion, and this is one reason why bounding the distortion of a given knot class is hard. I will show how to give a nontrivial lower bound on the distortion of torus knots, which is sharp in the case of (p,p+1) torus knots. I will also mention some natural conjectures about the distortion, for example that the distortion of the (2,p)-torus knots is unbounded.

Series: Geometry Topology Seminar

Caratheodory's famous conjecture, dating back to 1920's, states that every closed convex surface has at least two umbilics, i.e., points where the principal curvatures are equal, or, equivalently, the surface has contact of order 2 with a sphere. In this talk I report on recent work with Ralph howard where we apply the divergence theorem to obtain integral equalities which establish some weak forms of the conjecture.

Series: Geometry Topology Seminar

I will discuss a new general framework for cutting and gluing manifolds in topological quantum field theory (TQFT). Applying this method to Chern-Simons theory with gauge group SL(2,C) on a knot complement M leads to a systematic quantization of the SL(2,C) character variety of M. In particular, the classical A-polynomial of M becomes an operator "A-hat", the same operator that appears in the recursion relations of Garoufalidis et al. for colored Jones polynomials.

Series: Geometry Topology Seminar

We describe sufficient conditions which guarantee that a finite set of
mapping classes generate a right-angled Artin subgroup
quasi-isometrically embedded in the mapping class group. Moreover,
under these conditions, the orbit map to Teichmuller space is a
quasi-isometric embedding for both of the standard metrics. This is
joint work with Chris Leininger and Johanna Mangahas.

Series: Geometry Topology Seminar

We will discuss properties of manifolds obtained by deleting a totally geodesic ``divisor'' from hyperbolic manifold.
Fundamental groups of these manifolds do not generally fit into any class of groups studied by the geometric group theory, yet the groups turn out to be relatively hyperbolic when the divisor is ``sparse'' and has ``normal crossings''.

Series: Geometry Topology Seminar

Knotted trivalent graphs (KTGs) along with standard operations
defined on them form a finitely presented algebraic structure which
includes knots, and in which many topological knot properties are
defineable using simple formulas. Thus, a homomorphic invariant of KTGs
places knot theory in an algebraic context. In this talk we construct such
an invariant: the starting point is extending the Kontsevich integral of
knots to KTGs. This was first done in a series of papers by Le, Murakami,
Murakami and Ohtsuki in the late 90's using the theory of associators. We
present an elementary construction building on Kontsevich's original
definition, and discuss the homomorphic properties of the invariant,
which, as it turns out, intertwines all the standard KTG operations except
for one, called the edge unzip. We prove that in fact no universal finite
type invariant of KTGs can intertwine all the standard operations at once,
and present an alternative construction of the space of KTGs on which a
homomorphic universal finite type invariant exists. This space retains all
the good properties of the original KTGs: it is finitely presented,
includes knots, and is closely related to Drinfel'd associators. (Partly
joint work with Dror Bar-Natan.)

Series: Geometry Topology Seminar

Topological quantum field theory associates to a surface a sequence of
vector spaces and to curves on the surface, sequence of operators on
that spaces. It is expected that these operators are Toeplitz although
there is no general proof. I will state it in some particular cases and
give applications to the asymptotics of quantum invariants like quantum
6-j symbols or quantum invariants of Dehn fillings of the figure eight
knot. This is work in progress with (independently) L. Charles and T.
Paul.