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Series: Geometry Topology Seminar

A smooth loop in 3-space is skew if it has no pair of parallel tangent lines. With M.~Ghomi, we proved some years ago that among surfaces with some positive Gauss curvature (i.e., local convexity) the absence of skewloops characterizes quadrics. The relationship between skewloops and negatively curved surfaces has proven harder to analyze, however. We report some recent progress on that problem, including evidence both for and against the possibility that the absence of skewloops characterizes quadricsamong surfaces with negative curvature.

Series: Geometry Topology Seminar

I plan to discuss a method for defining Heegaard Floer invariants for 3-manifolds. The construction is inspired by contact geometry and has several interesting immediate applications to the study of tight contact structures on noncompact 3-manifolds. In this talk, I'll focus on one basic examples and indicate how one defines a contact invariant which can be used to give an alternate proof of James Tripp's classification of tight, minimally twisting contact structures on the open solid torus. This is joint work with John B. Etnyre and Rumen Zarev.

Series: Geometry Topology Seminar

We will discuss the structure of the symmetric (or hyperelliptic) Torelli group. More specifically, we will investigatethe group generated by Dehn twists about symmetric separating curvesdenoted by H(S). We will show that Aut(H(S)) is isomorphic to the symmetricmapping class group up to the hyperelliptic involution. We will do this bylooking at the natural action of H(S) on the symmetric separating curvecomplex and by giving an algebraic characterization of Dehn twists aboutsymmetric separating curves.

Series: Geometry Topology Seminar

A theorem of Chris Wendl allows you to completely characterize symplectic fillings of certain open book decompositions by factorizations of their monodromy into Dehn twists. Olga Plamenevskaya and I use this to generalize results of Eliashberg, McDuff and Lisca to classify the fillings of certain Lens spaces. I'll discuss this and a newer version of Wendl's theorem, joint with Wendl and Sam Lisi, this time for spinal open books, and discuss a few more applications.

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.