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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.

Series: Geometry Topology Seminar

This talk is about the dilatations of pseudo-Anosov mapping classes obtained by pushing a marked point around a filling curve. After reviewing this "point-pushing" construction, I will give both upper and lower bounds on the dilatation in terms of the self-intersection number of the filling curve. I'll also give bounds on the least dilatation of any pseudo-Anosov in the point-pushing subgroup and describe the asymptotic dependence on self-intersection number. All of the upper bounds involve analyzing explicit examples using train tracks, and the lower bound is obtained by lifting to the universal cover and studying the images of simple closed curves.

Series: Geometry Topology Seminar

This is the first talk in the Emory-Ga Tech-UGA joint seminar. The second talk will begin at 5:00. (NOTE: These talks are on the UGA campus.)

I will survey the program of realizing various versions of Floer homology as a theory of geometric cycles. This involves the description of infinite dimensional manifolds mapping to the relevant configuration spaces. This approach, which goes back to Atiyah's address at the Herman Weyl symposium, is in some ways technically simpler than the traditional construction based on Floer's version of Morse theory. In addition, it opens up the possibility of defining more refined invariants such as bordism andK-theory.

Series: Geometry Topology Seminar

We prove a conjecture of K. Schmidt in algebraic dynamical system theory onthe growth of the number of components of fixed point sets. We also prove arelated conjecture of Silver and Williams on the growth of homology torsions offinite abelian covering of link complements. In both cases, the growth isexpressed by the Mahler measure of the first non-zero Alexander polynomial ofthe corresponding modules. In the case of non-ablian covering, the growth of torsion is less thanor equal to the hyperbolic volume (or Gromov norm) of the knot complement.

Series: Geometry Topology Seminar

The talk is 1.5-2 hours long, and although some knowledge of HeegaardFloer homology and contact manifolds is useful I will spend some time inthe begining to review the basic notions. So the talk should be accessibleto everyone.

The first hour of this talk gives a gentle introduction to yet another version of Heegaard Floer homology; Sutured Floer homology. This is the generalization of Heegaard Floer homology, for 3-manifolds with decorations (sutures) on their boundary. Sutures come naturally for contact 3-manifolds. Later we will concentrate on invariants for contact 3--manifolds in Heegaard Floer homology. This can be defined both for closed 3--manifolds, in this case they live in Heegaard Floer homology and for 3--manifolds with boundary, when the invariant is in sutured Floer homology. There are two natural generalizations of these invariants for Legendrain knots. One can directly generalize the definition of the contact invariant $\widehat{\mathcal{L}}$, or one can take the complement of the knot, and compute the invariant for that:$\textrm{EH}$. At the end of this talk I would like to describe a map that sends $\textrm{EH}$ to$\widehat{\mathcal{L}}$. This is a joint work with Andr\'as Stipsicz.

Series: Geometry Topology Seminar

(joint work with M. Macasieb) Let $K$ be a hyperbolic $(-2, 3, n)$ pretzel knot and $M = S^3 \setminus K$ its complement. For these knots, we verify a conjecture of Reid and Walsh: there are at most three knotcomplements in the commensurability class of $M$. Indeed, if $n \neq 7$, weshow that $M$ is the unique knot complement in its class.

Series: Geometry Topology Seminar

Pseudo-Anosov mapping classes on surfaces have a rich structure, uncovered by William Thurston in the 1980's. We will discuss the 1995 Bestvina-Handel algorithmic proof of Thurston's theorem, and in particular the "transition matrix" T that their algorithm computes. We study the Bestvina-Handel proof carefully, and show that the dilatation is the largest real root of a particular polynomial divisor P(x) of the characteristic polynomial C(x) = | xI-T |. While C(x) is in general not an invariant of the mapping class, we prove that P(x) is. The polynomial P(x) contains the minimum polynomial M(x) of the dilatation as a divisor, however it does not in general coincide with M(x).In this talk we will review the background and describe the mathematics that underlies the new invariant. This represents joint work with Peter Brinkmann and Keiko Kawamuro.

Series: Geometry Topology Seminar

Series: Geometry Topology Seminar

Given a knot, a simple Lie algebra L and an irreducible representation V of L one can construct a one-variable polynomial with integer coefficients. When L is the simplest simple Lie algebra (sl_2) this gives a sequence of polynomials, whose sequence of degrees is a quadratic quasi-polynomial. We will discuss a conjecture for the degree of the colored Jones polynomial for an arbitrary simple Lie algebra, and we will give evidence for sl_3. This is joint work with Thao Vuong.