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A survey of the 3-manifold invariants derived from Hopf objects focusing on diagrammatic language.
by
Fernando J. O. de Souza
Los Alamos National Laboratory/ University of IL at Chicago
The research in quantum topology led to the construction of several 3-manifold invariants by means of either some kinds of Hopf algebras, or their categories of representations, or Hopf objects (objects in categories with reasonable structures, endowed with morphisms that satisfy the Hopf axioms as in the case of Hopf algebras.) This survey explores these invariants, particularly the involutory Kuperberg invariant and the Kauffman-Radford reformulation of the Hennings (KRH) invariant.
On the one hand, they can be seen as particular cases of other invariants when they are defined via some Hopf algebras. On the other hand, they can be defined at the general level of Hopf objects by means of diagrams representing morphisms. We will start by reviewing the effect of several categorical structures on the typical representation of objects (resp. morphisms) as edges (resp. vertices) of planar immersions of graphs. We will also demonstrate the use of diagrams to represent morphisms of Hopf objects modulo Hopf axioms, and recall the realization of this diagrams in Hopf algebras. We will then build the involutory Kuperberg and KRH invariants, and review their relations with other invariants (Witten-Reshetikhin-Turaev, Lyubashenko, Turaev-Viro, Chung-Fukuma-Shapere, and Barrett-Westbury) and some special cases. We will also cover: the completeness of the (diagrammatic) involutory Kuperberg invariant for prime, orientable, closed 3-manifolds; and the KRH invariant on a diagrammatic, Drinfeld quantum double of any involutory Hopf object (conjectured to be equivalent to the involutory Kuperberg invariant.) Finally, we will mention their generalizations to framed 3-manifolds due to Kuperberg and Sawin respectively.
Date received: February 4, 2000
Copyright © 2000 by the author(s). The author(s) of this work and the organizers of the conference have granted their consent to include this abstract in Topology Atlas. Document # caea-22.