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Organizers |
Continuation of plenary talk: Manifold pairings.
by
Michael Freedman
Microsoft Research
I'll go into more mathematical detail on the 'manifold pairings' theorem discussed in the plenary talk.
Date received: October 30, 2007
Foliations and non-metrisable manifolds
by
David Gauld
University of Auckland
Coauthors: Mathieu Baillif, Alexander Gabard, Paul Gartside and Sina Greenwood
Declare a topological space to be a manifold provided that it is connected and locally Euclidean. Manifolds can be non-metrisable for the simple reason that they are not Hausdorff or because they are too big.
Non-Hausdorff 1-manifolds are important in foliation theory because the leaf space of a foliation of the open disc in the plane is such a manifold. I shall describe a 1-manifold which is rigid in the sense that the only homeomorphism of it is the identity. This manifold gives rise to a foliation of the disc such that any leaf-respecting homeomorphism sends each leaf to itself.
Hausdorff manifolds that are too big to be metrisable are also interesting when it comes to foliations. For example there is a surface which carries no co-dimension 1 foliation; there is a 3-manifold with a co-dimension 1 foliation having only one leaf which necessarily is not metrisable. This last situation cannot occur in the metrisable case, nor in the non-metrisable case if the leaves are of dimension 1; in either case there are always continuum many leaves.
Date received: October 29, 2007
Covering spaces and the Kontsevich integral.
by
Andrew Kricker
Nanyang Technological University
Despite some remarkable insights in recent years, the question of how the quantum topology of knots and three-manifolds relates to their geometric topology remains, to a large extent, a fairly mysterious thing. Covering spaces are a setting in which this relationship can be productively explored. In this talk I'll describe some strong results about how quantum topology sees cyclic covers of knot complements (joint work with Stavros Garoufalidis), and also some preliminary results regarding another important class of covering spaces - the dihedral covers (joint work with Daniel Moskovitch).
Date received: October 4, 2007
Lasagna composition of Khovanov link homologies, and a 4-d skein module.
by
Scott Morrison
Microsoft Station Q
Coauthors: Kevin Walker
I'll describe a family of operations on the Khovanov homologies of links in S3, called `lasagna composition'. These are higher dimensional analogues of the operations in a planar algebra, or tensor category with duals. (In fact, you can think of this result as describing a braid tensor 2-category.) With these operations, we can define an invariant of a pair (W4, L ⊂ ∂W) (a link in the boundary of a 4-manifold), which recovers Khovanov's construction for (B4, L ⊂ S3).
Date received: September 21, 2007
The Ribbon Half-Twist
by
Noah Snyder
UC Berkeley
Coauthors: Peter Tingley
The theory of ribbon categories has an annoying defect, namely that you can only talk about a full 360-degree ribbon twist, but not a 180-degree half-twist. Building on recent work of Kamnitzer and Tingley, I'll explain how to interpret a half-twist. In particular, there is a beautiful picture which gives a formula for the braiding in terms of the half-twist.
Date received: October 31, 2007
Generalised knot groups
by
Christopher Tuffley
Massey University
Wada and Kelly independently introduced a family of knot invariants Gn(K) that generalise the fundamental group of a knot. The group G1(K) is the fundamental group, and Gn(K) is obtained by adjoining an nth root of the meridian that commutes with the longitude. I'll show that the isomorphism type of Gn(K), n ≥ 2, is a strictly stronger invariant of K than the isomorphism type of the fundamental group, by showing that the generalised knot groups of the square and granny knots are non-isomorphic for each n ≥ 2.
Date received: October 11, 2007
The Cyclotomic Birman-Murakami-Wenzl Algebras
by
Shona Yu
The University of Sydney
Coauthors: Stewart Wilcox
The Birman-Murakami-Wenzl (BMW) algebras are closely tied with the Artin braid group of type A, the Iwahori-Hecke algebras of type A (the symmetric group), the Brauer algebras and even quantum groups. Its algebraic definition was originally motivated by the Kauffman link invariant and, geometrically, it is isomorphic to the Kauffman tangle algebra. The representations and the cellularity of the BMW algebra have now been extensively studied in the literature.
Motivated by type B knot theory and the Ariki-Koike algebras (aka cyclotomic Hecke algebras of type G(r, 1, n)), Häring-Oldenburg defined the cyclotomic BMW algebras. In this talk, we present results regarding the structure of these algebras and give a geometric realization of the cyclotomic BMW algebras (in terms of "cylindrical" tangles). It turns out these algebras are also cellular, thereby allowing us to deduce information about its representations using Graham and Lehrer's general theory of cellular algebras.
Date received: October 24, 2007