426 research outputs found
HOMFLYPT Skein Theory, String Topology and 2-Categories
We show that relations in Homflypt type skein theory of an oriented
-manifold are induced from a -groupoid defined from the fundamental
-groupoid of a space of singular links in . The module relations are
defined by homomorphisms related to string topology. They appear from a
representation of the groupoid into free modules on a set of model objects. The
construction on the fundamental -groupoid is defined by the singularity
stratification and relates Vassiliev and skein theory. Several explicit
properties are discussed, and some implications for skein modules are derived.Comment: 55 pages, 1 figur
Deformation of string topology into homotopy skein modules
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.Comment: Published by Algebraic and Geometric Topology at
http://www.maths.warwick.ac.uk/agt/AGTVol3/agt-3-34.abs.htm
On Constructions of Generalized Skein Modules
Jozef Przytycki introduced skein modules of 3-manifolds and skein deformation initiating algebraic topology based on knots. We discuss the generalized skein modules of Walker, defined by fields and local relations. Some results by Przytycki are proven in a more general setting of fields defined by decorated cell-complexes in manifolds. A construction of skein theory from embedded TQFT-functors is given, and the corresponding background is developed. The possible coloring of fields by elements of TQFT-modules is discussed for those generalized skein modules. Also an approach of defining skein modules from studying compressions of fields is described
Frobenius algebras and skein modules of surfaces in 3-manifolds
For each 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.Comment: 24 page
Homflypt Skein Theory, String Topology and 2-Categories
We show that relations in Homflypt type skein theory of an oriented 3-manifold M are induced from a 2-groupoid defined from the fundamental 2-groupoid of a space of singular links M. The module relations are defined by homomorphisms related to string topology. They appear from a representation of the groupoid into free modules on a set of model objects. The construction on the fundamental 2-groupoid is defined by the singularity stratification and relates Vassiliev and skein theory. Several explicit properties are discussed, and some implications for skein modules are derived
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