The 2-twist spun trefoil is an example of a sphere that is knotted in 4-dimensional space. Here this example is shown to be distinct from the same sphere with the reversed orientation. To demonstrate this fact a state-sum 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 3-space. Colorings of diagrams of knotted curves and surfaces by quandle elements, together with cocycles of quandles, are used to define state-sum invariants for knotted circles in 3-space and knotted surfaces in 4-space. 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. Non-triviality of the invariants are proved for variety of knots and links, including the trefoil and figure-eight knots, and conversely, knot invariants are used to prove non-triviality of cohomology for a variety of quandles. 1 1

State-sum 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 4-space, movie methods and surface braid theory are used. Relations between the invariants and symmetries of knots are discussed.

State-sum 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 non-invertible 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 (self-distributive 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 state-sum invariants, defined diagrammatically using knot diagrams and quandle cocycles, for both classical knots in 3-space and knotted surfaces in 4-space. The invariant is used to give a proof that some 2-twist spun torus knots are noninvertible (not equivalent...

In this paper we define a class of state-sum invariants of compact closed oriented piece-wise linear 4-manifolds using finite groups. The definition of these state-sums follows from the general abstract construction of 4-manifold invariants using spherical 2-categories, as we defined in [32], although it requires a slight generalization of that construction. We show that the state-sum invariants of Birmingham and Rakowski [11, 12, 13], who studied Dijkgraaf-Witten 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 3-types. This connection suggests that there are many more interesting examples of our construction to be found in the work on homotopy 3-types, such as [15], for example. 1 1

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.

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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 space-time at the Plank scale; 2) a microscopical derivation of the Bekenstein-Hawking black-hole entropy formula. Unfortunately, despite recent results, the dynamical aspect of the theory (imposition of the Wheller-De 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 an...

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 M-theory. 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.

The 2-twist spun trefoil is an example of a sphere that is knotted in 4-dimensional space. Here this example is shown to be distinct from the same sphere with the reversed orientation. To demonstrate this fact a state-sum 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 3-space. Colorings of diagrams of knotted curves and surfaces by quandle elements, together with cocycles of quandles, are used to define state-sum invariants for knotted circles in 3-space and knotted surfaces in 4-space. 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. Non-triviality of the invariants are proved for variety of knots and links, including the trefoil and figure-eight knots, and conversely, knot invariants are used to prove non-triviality of cohomology for a variety of quandles. 1 1

We introduce a noncommutative and noncocommutative Hopf algebra HGT which takes for certain Hopf categories (and therefore braided monoidal bicategories) a similar role as the Grothendieck-Teichmüller group for quasitensor categories. We also give a result which highly restricts the possibility for similar structures for even higher weak n-categories than bicategories by showing that these structures would not allow for any nontrivial deformations. Finally, we suggest two possible constructions which might lead to representations of the Hopf algebra HGT. 1 The Hopf algebra HGT In [Dri] Drinfeld introduced the Grothendieck-Teichmüller group by considering the (formal) reparametrizations of the data (commutativity and associativity isomorphisms) of a quasitensor category. Consider now braided (weak) monoidal bicategories arising from the representations of a Hopf category