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Series: Geometry Topology Seminar

In this talk I will describe how the quantum determinant modelof the Colored Jones polynomial, developed by Vu Huynh and Thang Le can beinterpreted in a combinatorial way as walks along a braid. Thisinterpretation can then be used to prove that the leading coefficients ofthe colored Jones polynomial stabalize, defining two power series calledthe head and the tail. I will also show examples where the head and tailcan be calculated explicitly and have applications in number theory.

Series: Geometry Topology Seminar

I'll describe a new combinatorial method for computing the delta-graded knot Floer homology of a link in S^3. Our construction comes from iterating an unoriented skein exact triangle discovered by Manolescu, and yields a chain complex for knot Floer homology which is reminiscent of that of Khovanov homology, but is generated (roughly) by spanning trees of the black graph of the link. This is joint work with Adam Levine.

Series: Geometry Topology Seminar

The actual talk will be 40 minutes. Note the unusual time.

The theorem of Birman and Hilden relates the mapping class group of a surface and its image under a covering map. I'll explore when we can extend the original theorem and possible applications for further work.

Series: Geometry Topology Seminar

I will consider two constructions which lead to information about the topology of a 3-manifold from one of its triangulation. The first construction is a modification of the Turaev-Viro invariant based on re-normalized 6j-symbols. These re-normalized 6j-symbols satisfy tetrahedral symmetries. The second construction is a generalization of Kashaev's invariant defined in his foundational paper where he first stated the volume conjecture. This generalization is based on symmetrizing 6j-symbols using *charges* developed by W. Neumann, S. Baseilhac, and R. Benedetti. In this talk, I will focus on the example of nilpotent representations of quantized sl(2) at a root of unity. In this example, the two constructions are equal and give rise to a kind of Homotopy Quantum Field Theory. This is joint work with R. Kashaev, B. Patureau and V. Turaev.

Series: Geometry Topology Seminar

I will talk about some progress in proving the Degree Conjecture for torus
knots. The conjecture states that the degree of a colored Jones polynomial colored by an
irreducible representation of a simple Lie algebra g is locally a quadratic
quasi-polynomial. This is joint work with Stavros Garoufalidis.

Series: Geometry Topology Seminar

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.