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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 65 (16 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.
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 20 (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
On Second Quantization of Quantum Groups
, 2000
"... We construct a deformation of the function algebra on the quantum group SL q (2) into a trialgebra in the sense of Crane and Frenkel. We show that this naturally acts on the trialgebraic deformation of the Manin plane, previously introduced by the authors. Alternatively, one can view it as acting o ..."
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Cited by 7 (1 self)
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We construct a deformation of the function algebra on the quantum group SL q (2) into a trialgebra in the sense of Crane and Frenkel. We show that this naturally acts on the trialgebraic deformation of the Manin plane, previously introduced by the authors. Alternatively, one can view it as acting on the trialgebraic deformation of the fermionic Manin plane. We prove that the trialgebraic deformation of SL q (2) defines a 2  C*category, a structure as needed for the superselection structure of massive two dimensional quantum field theories. Besides this, we investigate another approach to trialgebra deformations of a bialgebra as a deformation of a Fock space construction over the bialgebra.
Canonical ”loop” quantum gravity and spin foam models”, to appear in the proceedings of the XXIIIth Congress of the Italian Society for General Relativity and
 Gravitational Physics (SIGRAV
, 1998
"... The canonical “loop ” formulation of quantum gravity is a mathematically well defined, background independent, non perturbative standard quantization of Einstein’s theory of General Relativity. Some among the most meaningful results of the theory are: 1) the complete calculation of the spectrum of g ..."
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Cited by 4 (1 self)
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The canonical “loop ” formulation of quantum gravity is a mathematically well defined, background independent, non perturbative standard quantization of Einstein’s theory of General Relativity. Some among the most meaningful results of the theory are: 1) the complete calculation of the spectrum of geometric quantities like the area and the volume and the consequent physical predictions about the structure of the spacetime at the Planck scale; 2) a microscopical derivation of the BekensteinHawking blackhole entropy formula. Unfortunately, despite recent results, the dynamical aspect of the theory (imposition of the WhellerDe Witt constraint) remains elusive. After a short description of the basic ideas and the main results of loop quantum gravity we show in which sence the exponential of the super Hamiltonian constraint leads to the concept of spin foam and to a four dimensional formulation of the theory. Moreover, we show that some topological field theories as the BF theory in 3 and 4 dimension admits a spin foam formulation. We argue that the spinfoams/spinnetworks formalism it is the natural framework to discuss loop quantum gravity and topological field theory. CPT99/P.3796
A PREHISTORY OF nCATEGORICAL PHYSICS
, 2008
"... We begin with a chronology tracing the rise of symmetry concepts in physics, starting with groups and their role in relativity, and leading up to more sophisticated concepts from ncategory theory, which manifest themselves in Feynman diagrams and their higherdimensional generalizations: strings, me ..."
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Cited by 1 (1 self)
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We begin with a chronology tracing the rise of symmetry concepts in physics, starting with groups and their role in relativity, and leading up to more sophisticated concepts from ncategory theory, which manifest themselves in Feynman diagrams and their higherdimensional generalizations: strings, membranes and spin foams.
Deformations of Conformal Field Theories to Models with Noncommutative World Sheets
, 2000
"... We discuss deformations of conformal eld theories to bosonic models where both  the target space and the world sheet  become noncommutative spaces (in the sense of noncommutative geometry). We give an action for such models which in the commutative limit is on the classical level equivalent to the ..."
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We discuss deformations of conformal eld theories to bosonic models where both  the target space and the world sheet  become noncommutative spaces (in the sense of noncommutative geometry). We give an action for such models which in the commutative limit is on the classical level equivalent to the usual string action. When a special choice of noncommutative target is made and for the world sheet a kind of point particle limit is taken, one gets the bosonic part of the action of the BFSS matrix model of Mtheory. Besides this, we briey discuss the question of the underlying symmetries (comparable to the quantum group symmetries of usual two dimensional conformal eld theories) and argue that trialgebraic deformations of Hopf algebras should appear, here. A brief look at dualities concludes the discussion.
Dedicated to Professor Kunio Murasugi for his 70th birthday
, 2008
"... 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 ..."
Abstract
 Add to MetaCart
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. 1 1