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Quandle cohomology and statesum invariants of knotted curves and surfaces
 TRANS. AMER. MATH. SOC
, 1999
"... The 2twist spun trefoil is an example of a sphere that is knotted in 4dimensional space. Here this example is shown to be distinct from the same sphere with the reversed orientation. To demonstrate this fact a statesum invariant for classical knots and knotted surfaces is developed via a cohomolo ..."
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Cited by 64 (17 self)
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The 2twist spun trefoil is an example of a sphere that is knotted in 4dimensional space. Here this example is shown to be distinct from the same sphere with the reversed orientation. To demonstrate this fact a statesum invariant for classical knots and knotted surfaces is developed via a cohomology theory of racks and quandles (also known as distributive groupoids). A quandle is a set with a binary operation — the axioms of which model the Reidemeister moves in the classical theory of knotted and linked curves in 3space. Colorings of diagrams of knotted curves and surfaces by quandle elements, together with cocycles of quandles, are used to define statesum invariants for knotted circles in 3space and knotted surfaces in 4space. Cohomology groups of various quandles are computed herein and applied to the study of the statesum invariants of classical knots and links and other linked surfaces. Nontriviality of the invariants are proved for variety of knots and links, including the trefoil and figureeight knots, and conversely, knot invariants are used to prove nontriviality of cohomology for a variety of quandles.
An invariant of link cobordisms from Khovanov’s homology theory
 Algebr. Geom. Topol
"... 1.1. Khovanov’s Homology. In [K] M.Khovanov introduced a new homology theory, which assigns to a diagram D of an oriented classical link L a bigraded family of homology groups Hi,j (D) such that the graded Euler characteristic ∑ ..."
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Cited by 51 (1 self)
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1.1. Khovanov’s Homology. In [K] M.Khovanov introduced a new homology theory, which assigns to a diagram D of an oriented classical link L a bigraded family of homology groups Hi,j (D) such that the graded Euler characteristic ∑
Higherdimensional algebra VI: Lie 2algebras,
, 2004
"... The theory of Lie algebras can be categorified starting from a new notion of ‘2vector space’, which we define as an internal category in Vect. There is a 2category 2Vect having these 2vector spaces as objects, ‘linear functors’ as morphisms and ‘linear natural transformations ’ as 2morphisms. We ..."
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Cited by 43 (12 self)
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The theory of Lie algebras can be categorified starting from a new notion of ‘2vector space’, which we define as an internal category in Vect. There is a 2category 2Vect having these 2vector spaces as objects, ‘linear functors’ as morphisms and ‘linear natural transformations ’ as 2morphisms. We define a ‘semistrict Lie 2algebra ’ to be a 2vector space L equipped with a skewsymmetric bilinear functor [·, ·]: L × L → L satisfying the Jacobi identity up to a completely antisymmetric trilinear natural transformation called the ‘Jacobiator’, which in turn must satisfy a certain law of its own. This law is closely related to the Zamolodchikov tetrahedron equation, and indeed we prove that any semistrict Lie 2algebra gives a solution of this equation, just as any Lie algebra gives a solution of the Yang–Baxter equation. We construct a 2category of semistrict Lie 2algebras and prove that it is 2equivalent to the 2category of 2term L∞algebras in the sense of Stasheff. We also study strict and skeletal Lie 2algebras, obtaining the former from strict Lie 2groups and using the latter to classify Lie 2algebras in terms of 3rd cohomology classes in Lie algebra cohomology. This classification allows us to construct for any finitedimensional Lie algebra g a canonical 1parameter family of Lie 2algebras g � which reduces to g at � = 0. These are closely related to the 2groups G � constructed in a companion paper.
An invariant of tangle cobordisms
"... In [9] to a plane diagram D of an oriented tangle T with 2n bottom and 2m top endpoints we associated a complex F(D) of (H m, H n)bimodules, ..."
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Cited by 30 (3 self)
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In [9] to a plane diagram D of an oriented tangle T with 2n bottom and 2m top endpoints we associated a complex F(D) of (H m, H n)bimodules,
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...
Statesum Invariants of Knotted Curves and Surfaces from Quandle Cohomology
 Electronic Research Announcements of the AMS
, 1999
"... Statesum invariants for classical knots and knotted surfaces in 4space are developed via the cohomology theory of quandles. Cohomology of quandles are computed to evaluate the invariants. Some twist spun torus knots are shown to be noninvertible using the invariants. 1 1 Introduction The purpose ..."
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Cited by 19 (2 self)
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Statesum invariants for classical knots and knotted surfaces in 4space are developed via the cohomology theory of quandles. Cohomology of quandles are computed to evaluate the invariants. Some twist spun torus knots are shown to be noninvertible using the invariants. 1 1 Introduction The purpose of this paper is to present a summary of a series of papers [4, 6, 7]. It is based on the research announcement [5] but has been expanded to include recent developments. A cohomology theory for racks (selfdistributive groupoids, defined below) was defined and the general framework for defining invariants of codimension 2 embeddedings was outlined in [14] and [15] from an algebrotopological view point. The present paper announces statesum invariants, defined diagrammatically using knot diagrams and quandle cocycles, for both classical knots in 3space and knotted surfaces in 4space. The invariant is used to give a proof that some 2twist spun torus knots are noninvertible (not equivalent...
Computations of quandle cocycle invariants of knotted curves and surfaces
, 1999
"... Statesum invariants for knotted curves and surfaces using quandle cohomology were introduced by Laurel Langford and the authors in [4]. In this paper we present methods to compute the invariants and sample computations. Computer calculations of cohomological dimensions for some quandles are present ..."
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Cited by 17 (6 self)
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Statesum invariants for knotted curves and surfaces using quandle cohomology were introduced by Laurel Langford and the authors in [4]. In this paper we present methods to compute the invariants and sample computations. Computer calculations of cohomological dimensions for some quandles are presented. For classical knots, Burau representations together with Maple programs are used to evaluate the invariants for knot table. For knotted surfaces in 4space, movie methods and surface braid theory are used. Relations between the invariants and symmetries of knots are discussed.
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
Categorification
 Contemporary Mathematics 230. American Mathematical Society
, 1997
"... Categorification is the process of finding categorytheoretic analogs of settheoretic concepts by replacing sets with categories, functions with functors, and equations between functions by natural isomorphisms between functors, which in turn should satisfy certain equations of their own, called ‘c ..."
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Cited by 4 (1 self)
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Categorification is the process of finding categorytheoretic analogs of settheoretic concepts by replacing sets with categories, functions with functors, and equations between functions by natural isomorphisms between functors, which in turn should satisfy certain equations of their own, called ‘coherence laws’. Iterating this process requires a theory of ‘ncategories’, algebraic structures having objects, morphisms between objects, 2morphisms between morphisms and so on up to nmorphisms. After a brief introduction to ncategories and their relation to homotopy theory, we discuss algebraic structures that can be seen as iterated categorifications of the natural numbers and integers. These include tangle ncategories, cobordism ncategories, and the homotopy ntypes of the loop spaces Ω k S k. We conclude by describing a definition of weak ncategories based on the theory of operads. 1