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20
Higherdimensional algebra IV: 2Tangles
"... Just as knots and links can be algebraically described as certain morphisms in the category of tangles in 3 dimensions, compact surfaces smoothly embedded in R 4 can be described as certain 2morphisms in the 2category of ‘2tangles in 4 dimensions’. Using the work of Carter, Rieger and Saito, we p ..."
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Cited by 35 (10 self)
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Just as knots and links can be algebraically described as certain morphisms in the category of tangles in 3 dimensions, compact surfaces smoothly embedded in R 4 can be described as certain 2morphisms in the 2category of ‘2tangles in 4 dimensions’. Using the work of Carter, Rieger and Saito, we prove that this 2category is the ‘free semistrict braided monoidal 2category with duals on one unframed selfdual object’. By this universal property, any unframed selfdual object in a braided monoidal 2category with duals determines an invariant of 2tangles in 4 dimensions. 1
Structures and Diagrammatics of Four Dimensional Topological Lattice Field Theories
 Advances in Math. 146
, 1998
"... Crane and Frenkel proposed a state sum invariant for triangulated 4manifolds. They defined and used new algebraic structures called Hopf categories for their construction. Crane and Yetter studied Hopf categories and gave some examples using group cocycles that are associated to the Drinfeld double ..."
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Cited by 20 (5 self)
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Crane and Frenkel proposed a state sum invariant for triangulated 4manifolds. They defined and used new algebraic structures called Hopf categories for their construction. Crane and Yetter studied Hopf categories and gave some examples using group cocycles that are associated to the Drinfeld double of a finite group. In this paper we define a state sum invariant of triangulated 4manifolds using CraneYetter cocycles as Boltzmann weights. Our invariant generalizes the 3dimensional invariants defined by Dijkgraaf and Witten and the invariants that are defined via Hopf algebras. We present diagrammatic methods for the study of such invariants that illustrate connections between Hopf categories and moves to triangulations. 1 Contents 1 Introduction 3 2 Quantum 2 and 3 manifold invariants 4 Topological lattice field theories in dimension 2 . . . . . . . . . . . . . . . . . . . 4 Pachner moves in dimension 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 TuraevViro inv...
Finite groups, spherical 2categories, and 4manifold invariants. arXiv:math.QA/9903003
"... In this paper we define a class of statesum invariants of compact closed oriented piecewise linear 4manifolds using finite groups. The definition of these statesums follows from the general abstract construction of 4manifold invariants using spherical 2categories, as we defined in [32], althou ..."
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Cited by 16 (5 self)
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In this paper we define a class of statesum invariants of compact closed oriented piecewise linear 4manifolds using finite groups. The definition of these statesums follows from the general abstract construction of 4manifold invariants using spherical 2categories, as we defined in [32], although it requires a slight generalization of that construction. We show that the statesum invariants of Birmingham and Rakowski [11, 12, 13], who studied DijkgraafWitten type invariants in dimension 4, are special examples of the general construction that we present in this paper. They showed that their invariants are nontrivial by some explicit computations, so our construction includes interesting examples already. Finally, we indicate how our construction is related to homotopy 3types. This connection suggests that there are many more interesting examples of our construction to be found in the work on homotopy 3types, such as [15], for example. 1 1
A functorvalued invariant of tangles Contents
, 2001
"... 2 A bimodule realization of the TemperleyLieb twocategory 8 ..."
unknown title
, 2001
"... This paper gives a selfcontained and complete proof of the isomorphism of freely generated monoids extracted from TemperleyLieb algebras with monoids made of Kauffman’s diagrams. Mathematics Subject Classification (2000): 57M99, 20M05 0 ..."
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This paper gives a selfcontained and complete proof of the isomorphism of freely generated monoids extracted from TemperleyLieb algebras with monoids made of Kauffman’s diagrams. Mathematics Subject Classification (2000): 57M99, 20M05 0
unknown title
, 2001
"... This paper gives a selfcontained and complete proof of the isomorphism of freely generated monoids extracted from TemperleyLieb algebras with monoids made of Kauffman’s diagrams. Mathematics Subject Classification (2000): 57M99, 20M05 0 ..."
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This paper gives a selfcontained and complete proof of the isomorphism of freely generated monoids extracted from TemperleyLieb algebras with monoids made of Kauffman’s diagrams. Mathematics Subject Classification (2000): 57M99, 20M05 0
unknown title
, 2000
"... This paper gives a selfcontained and complete proof of the isomorphism of freely generated monoids extracted from TemperleyLieb algebras with monoids made of Kauffman’s diagrams. Mathematics Subject Classification (2000): 57M99, 20M05 0 ..."
Abstract
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This paper gives a selfcontained and complete proof of the isomorphism of freely generated monoids extracted from TemperleyLieb algebras with monoids made of Kauffman’s diagrams. Mathematics Subject Classification (2000): 57M99, 20M05 0
unknown title
, 2000
"... This paper gives a selfcontained and complete proof of the isomorphism of freely generated monoids extracted from TemperleyLieb algebras with monoids made of Kauffman’s diagrams. Mathematics Subject Classification (2000): 57M99, 20M05 0 ..."
Abstract
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This paper gives a selfcontained and complete proof of the isomorphism of freely generated monoids extracted from TemperleyLieb algebras with monoids made of Kauffman’s diagrams. Mathematics Subject Classification (2000): 57M99, 20M05 0