Schedule of Topology Seminars: 2011-12

Recently several teams of mathematicians were able to construct a manifold with two different Heegaard splittings so that their common stabilization had high genus. In this talk we will discuss the natural generalization of this question to bridge surfaces.

Let S be a closed oriented surface of negative Euler characteristic. We consider the space Rep_n(S) of conjugacy classes of homomorphisms from the fundamental group of S to PSL_n(R), with n > 2. Using Higgs bundle techniques, N. Hitchin described the number of connected components of Rep_n(S). In particular, he gave a parametrization of one connected component, called the Hitchin space, which contains a copy of the Teichmuller space of S. Given a closed curve c on S and a representation r in the Hitchin space of S, we can consider the eigenvalues of r(c). We first show how to extend these eigenvalue functions to length functions on the space of measure geodesic currents on S, or more generally on the space of Hšlder geodesic currents. Then we introduce cataclysm deformations for Hitchin representations, and study the effect of these deformations on the length functions of a Hitchin representation. This work is based on Labourie's dynamical characterization of Hitchin representations.

We give an introduction to the Kakimizu complex of a link, covering a number of recent results. In particular we will see that the Kakimizu complex of a knot may be locally infinite, that the Alexander polynomial of an alternating link carries information about its Seifert surfaces, and that the Kakimizu complex of a special alternating link is understood.

We introduce tangles of type En and relations on these tangles. Our tangles are the diagrammatic versions of the Birman-Murakami-Wenzl algebra elements of type En. We don't show any isomorphism between the new tangle algebra and the BMW algebra. We simply use the tangles as a tool to construct a representation of the Artin group of type E6. The representation that we build is equivalent to the faithful generalized Lawrence-Krammer representation. We explain how weÊget a reducibility criterion for the representation.ÊWe alsoÊderive complex values of the parameters for which the BMW algebra of type E6 is not semisimple.

Casson and Gordon showed that irreducible 3-manifolds with distance one Heegaard splittings are Haken. We consider 3-manifolds with distance two Heegaard splittings. These are well-understood in the case of genus two. When the genus is greater than two, we show they fall into a few groups, one of which is Haken manifolds. Another is those that can be constructed by gluing three handlebodies together (carefully) along their boundaries. Under some constraints we are able to show that the fundamental groups of the handlebodies inject into the fundamental group of the manifold.Ê

The Kauffman bracket skein module K(M) of a 3-manifold M is defined by Przytychi as an invariant for framed links in M satisfying the Kauffman skein relation. For a compact oriented surface S, it is shown by Bullock, Frohman and Kania Bartoszynska that K(S x [0, 1]) is a quantization of the SL2C-characters of the fun- damental group of S with respect to the Goldman-Weil-Petersson Poisson bracket. In a joint work with Julien Roger that I will be talking about, we define a skein algebra of a punctured surface as an invariant for not only framed links but also framed arcs in S x? [0, 1] satisfying the skein relations of crossings both in the surface and at punctures. This construction relies on a collection of geodesic lengths identities in hyperbolic geometry which generalize PennerÕs Ptolemy relation, the SL2 trace identity and WolpertÕs cosine formula; and the construction itself could be considered as an intermediate step toward the quantization of the decorated Teichmuller space with the Weil-Petersson Poisson structure.

A genus g surface gives rise to two interesting graphs: the pants complex, defined by Hatcher and Thurston, and the dual curve complex, defined by Jesse Johnson. ÊRecently, Johnson has used paths in the pants and dual curve complexes of Heegaard surfaces for a 3-manifold M to define invariants of M. ÊAnalogously, we will show how paths in the pants and dual curve complexes of bridge surfaces for a knot K can be used to produce invariants of K. ÊWe will compute some of these invariants for a class of knots and discuss intriguing connections between the pants complex and hyperbolic geometry.

We derive relations between the coefficients of colored Jones polynomials and the topology of incompressible spanning surfaces in knot and link diagrams, for A-adequate links. This allows us to detect fibers and bound volumes for large classes of knots and links. Our approach is to generalize the checkerboard surfaces of alternating knot decompositions, show these surfaces are incompressible, and obtain a polyhedral decomposition of their complement. This allows us to relate the geometry of the complement to spines of the checkerboard surface (state graphs), which in turn are related to coefficients of the Jones polynomial. This is joint with David Futer and Efstratia Kalfagianni.

ÊIn this talk, we introduce works on bridge presentations of links, including some works done by the speaker. Results on 3-bridge links and their 3-bridge presentations will be introduced in the first half of this talk. The last half will be devoted to introduce a joint work with Michel Boileau on the relation between bridge numbers of links and minimal numbers of meridian generators of link groups.

We construct links of arbitrarily high tunnel number which experience high degeneration under connect sum with good pairs of knots, drawing on a construction of Morimoto. We then discuss generalizations of Morimoto's construction and the possibility of using such constructions to create more examples of higher degeneration.