Uwe Kaiser

Deformation of String Topology into Homotopy Skein Modules

Uwe Kaiser — Wed, 25 Apr 2012 10:42:31 PDT

Relations between the string topology of Chas and Sullivan and the homotopy skein modules of Hoste and Przytycki are studied. This provides new insight into the structure of homotopy skein modules and their meaning in the framework of quantum topology. Our results can be considered as weak extensions to all orientable 3–manifolds of classical results by Turaev and Goldman concerning intersection and skein theory on oriented surfaces.

Quantum Deformations of Fundamental Groups of Oriented 3-Manifolds

Uwe Kaiser — Wed, 25 Apr 2012 10:42:29 PDT

We compute two-term skein modules of framed oriented links in oriented 3-manifolds. They contain the self-writhe and total linking number invariants of framed oriented links in a universal way. The relations in a natural presentation of the skein module are interpreted as monodromies in the space of immersions of circles into the 3-manifold.

Bar-Natan Modules and Bar-Natan Pairings of Oriented 3-Manifolds

Uwe Kaiser — Tue, 24 Apr 2012 14:58:44 PDT

We discuss sesquilinear pairings defined by Bar-Natan modules (and their generalizations using general Frobenius algebras), which descend from universal manifold pairings recently discussed by Calegari, Freedman, Walker and others. Such a Bar-Natan pairing exists for each oriented closed surface with an embedded oriented closed 1-manifold (and each Frobenius algebra with involution). We also discuss how the Heegaard genus of closed 3-manifolds naturally appears in the calculation of Bar-Natan modules, and more generally how the calculation of Bar-Natan modules is related with the geometric topology of the 3-manifold.

Commutative Frobenius Algebras and 3-Dimensional Compression Bordisms

Uwe Kaiser — Tue, 24 Apr 2012 14:58:43 PDT

Given a commutative Frobenius algebra V over a commutative ring R with 1 we construct certain V^j -module categories V [j] for j ≥ 0. Let (M, α) be an oriented 3-manifold with a closed oriented 1-manifold α in its boundary. Then there are defined natural functors from a category of oriented surfaces in M bounding α and morphisms defined by compression bordisms in M × I, taking values in V [|α|]. (Here a compression bordism S₁ → S₂ is a 3-dimensional manifold with corners, properly embedded in M × I, which is a product over α, and with only embedded 2-handles and 3-handles attached to S₁ × I, considered up to isotopy through those bordisms). The colimit of this functor is the Bar-Natan skein module defined for (M, α) and the Frobenius algebra V . Moreover, a glueing theorem can be proven for this functor. The above construction can be twisted with a (3 + 1)-dimensional TQFT over R to define functors on a category with the morphisms embedded in oriented 4-manifolds. We discuss the above constructions and some conjectures related to it.

Bar-Natan Modules and Tunneling Graphs

Uwe Kaiser — Tue, 24 Apr 2012 14:58:41 PDT

We describe a general method for presentations of colimit modules of functors into module categories. This is applied to the Bar-Natan functor, which is defined on a category of surfaces embedded in a 3-manifold M with morphisms defined by certain 3-manifolds embedded in M x [0,1] and takes values in a category of modules defined from a commutative Frobenius algebra. The colimit of the Bar-Natan functor is the Bar-Natan module of M. Our approach naturally leads to the definition of the tunneling graph of M, which contains the geometric data necessary to deduce the structure of the Bar-Natan module.

Link Theory in Manifolds

Uwe Kaiser — Tue, 24 Apr 2012 14:58:39 PDT

Any topological theory of knots and links should be based on simple ideas of intersection and linking. In this book, a general theory of link bordism in manifolds and universal constructions of linking numbers in oriented 3-manifolds are developed. In this way, classical concepts of link theory in the 3-spheres are generalized to a certain class of oriented 3-manifolds (submanifolds of rational homology 3-spheres). The techniques needed are described in the book but basic knowledge in topology and algebra is assumed. The book should be of interst to those working in topology, in particular knot theory and low-dimensional topology.

Link Homotopy in the 2-Metastable Range

Nathan Habegger et al. — Tue, 24 Apr 2012 14:58:37 PDT

Deformation of Homotopy into Isotopy in Oriented 3-Manifolds

Uwe Kaiser — Tue, 24 Apr 2012 14:58:35 PDT

We will show that deformation quantization in skein theory of oriented 3-manifolds is induced from a topological deformation quantization of the fundamental 2-groupoid of the space of immersions of circles in M. The structure of skein module and its relations with string topology homomorphisms appear through representations of the groupoid structure into the set the objects. The deformation of the fundamental 2-groupoid is defined by the singularity stratification, the quantization by passage to isotopy classes. Several explicit properties and computations of skein modules are proved. It will be shown that local systems on the space of immersions are important for the understanding of HOMFLY oriented and framed skein theory. The passage from Conway to Jones skein theory is described on the categorical level.

From Euler to Witten: A Short Survey of the Volume Conjecture in Knot Theory

Uwe Kaiser — Tue, 24 Apr 2012 14:58:33 PDT

Frobenius Algebras and Skein Modules of Surfaces in 3-Manifolds

Uwe Kaiser — Tue, 24 Apr 2012 14:58:31 PDT

For each (commutative) Frobenius algebra there is defined a skein module of surfaces embedded in a given 3-manifold and bounding a prescribed curve system in the boundary. The skein relations are local and generate the kernel of a certain natural extension of the corresponding topological quantum field theory. In particular the skein module of the 3-ball is isomorphic to the ground ring of the Frobenius algebra. We prove a presentation theorem for the skein module with generators incompressible surfaces colored by elements of a generating set of the Frobenius algebra, and with relations determined by tubing geometry in the manifold and relations of the algebra.

Link Bordism Skein Modules

Uwe Kaiser — Tue, 24 Apr 2012 14:58:26 PDT

We compute link bordism skein modules of colored oriented links in oriented 3-manifolds. A Hurewicz theorem relating link bordism and link homotopy skein modules is proved.

Presentations of Homotopy Skein Modules of Oriented 3-Manifolds

Uwe Kaiser — Tue, 24 Apr 2012 14:58:24 PDT

A new method to derive presentations of skein modules is developed. For the case of homotopy skein modules it will be shown how the topology of a 3-manifold is reflected in the structure of the module. The freeness problem for q-homotopy skein modules is solved, and a natural skein module related to linking numbers is computed.