Right integrals and invariants of three-manifolds
This paper gives a summary of our approach to invariants of three manifolds via right integrals on finite dimensional Hopf algebras and their relation to the Kirby calculus.
math.GT↗
arXiv subjects
Louis H. Kauffman
Publications and source records attributed to Louis H. Kauffman.
On Two Proofs for the Existence and Uniqueness of Integrals for Finite-Dimensional Hopf Algebras
This paper has two purposes. The first is to explicate the diagrammatic approach to Hopf algebras due to Kuperberg, and to examine his proof of the existence and uniqueness of integrals in both the diagrammatic and purely algebraic contexts. The second purpose of the paper is to show that the theory of integrals for a finite dimensional Hopf algebra A can be deduced from ideas concerning the trace function on End(A). Connections between Kuperberg's work and the trace function should be of interest to those who study three manifold invariants in relation to Hopf algebras.
math.QA↗
Functional Integration and the Kontsevich Integral
This paper is an exposition of heuristics related to Witten's functional integral, relating it to Vassiliev invariants and to the Kontsevich integrals that can be used to produce Vassiliev invariants of knots and links.In particular, we give a simplified version of the appearance of the Kontsevich integrals in the perturbation expansion of the functional integral.
math.GT↗
Structures and Diagrammatics of Four Dimensional Topological Lattice Field Theories
Crane and Frenkel proposed a state sum invariant for triangulated 4-manifolds.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 4-manifolds using Crane-Yetter cocycles as Boltzmann weights. Our invariant generalizes the 3-dimensional 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.
math.GT↗
Fourier Knots
This paper introduces the concept of a Fourier knot. A Fourier knot is a knot that is represented by a parametrized curve in three dimensional space such that the coordinate functions are finite Fourier series in the parameter. The previously studied Lissajous knots constitute a case of Fourier knots with single frequencies in each coordinate direction. Not all knots are Lissajous. In fact the trefoil knot and the figure eight knot are the first examples of non- Lissajous knots. This paper gives robust and simple Fourier representations for these and other examples. This new version of the paper contains a crucial reference to the beautiful work of Aaron Trautwein on Harmonic Knots. See . See also .
q-alg↗
Noncommutativity and Discrete Physics
The purpose of this paper is to present an introduction to a point of view for discrete foundations of physics. In taking a discrete stance, we find that the initial expression of physical theory must occur in a context of noncommutative algebra and noncommutative vector analysis. In this way the formalism of quantum mechanics occurs first, but not necessarily with the usual interpretations. The basis for this work is a non-commutative discrete calculus and the observation that it takes one tick of the discrete clock to measure momentum.
q-alg↗
On Finiteness of Certain Vassiliev Invariants
The best known examples of Vassiliev invariants are the coefficients of a Jones-type polynomial expanded after exponential substitution. We show that for a given knot, the first $N$ Vassiliev invariants in this family determine the rest for some integer $N$.
q-alg↗
State-Sum Invariants of 4-Manifolds I
We provide, with proofs, a complete description of the authors' construction of state-sum invariants announced in [CY], and its generalization to an arbitrary (artinian) semisimple tortile category. We also discuss the relationship of these invariants to generalizations of Broda's surgery invariants [Br1,Br2] using techniques developed in the case of the semi-simple sub-quotient of $Rep(U_q(sl_2))$ ($q$ a principal $4r^{th}$ root of unity) by Roberts [Ro1]. We briefly discuss the generalizations to invariants of 4-manifolds equipped with 2-dimensional (co)homology classes introduced by Yetter [Y6] and Roberts [Ro2], which are the subject of the sequel. (citations refer to bibliography in the paper)
hep-th↗
Invariants of 3-Manifolds Derived From Finite Dimensional Hopf Algebras
This paper studies invariants of 3-manifolds derived from certain fin ite dimensional Hopf algebras. The invariants are based on right integrals for these algebras. It is shown that the resulting class of invariants is distinct from the class of Witten-Reshetikhin-Turaev invariants.
hep-th↗
Vassiliev Invariants and the Loop States in Quantum Gravity
An exposition of Vassiliev invariants is given in terms of the simplest approach to the functional integral construction of link invariants from Chern-Simons theory. This approach gives the top row evaluations of Vassiliev invariants for the classical Lie algebras, and a neat point of view on the results of Bar-Natan. It also clarifies the relation between Vassiliev invariants and the extension of the bracket invariant to links with transverse double points that appears in the work of Bruegmann, Gambini and Pullin on the loop representation of quantum gravity. We see that the Vassiliev vertex is not just a transversal intersection of Wilson loops, but rather has the structure of Casimir insertion (up to first order of approximation) coming from the difference formula in the functional integral. (Figures are available from the author.)
gr-qc↗
On the Classicality of Broda's SU(2) Invariant of 4-manifolds
Recent work of Roberts has shown that the first surgical 4-manifold invariant of Broda and (up to an unspecified normalization factor) the state-sum invariant arising from the TQFT of Crane-Yetter are equivalent to the signature of the 4-manifold. Subsequently Broda defined another surgical invariant in which the 1- and 2-handles are treated differently. We use a refinement of Roberts' techniques developped by the authors in hep-th/9309063 to show that the "improved" surgical invariant of Broda also depends only on the signature and Euler character
hep-th↗
Evaluating the Crane-Yetter Invariant
We provide an explicit formula for the invariant of 4-manifolds introduced by Crane and Yetter (in hep-th 9301062). A consequence of our result is the existence of a combinatorial formula for the signature of a 4-manifold in terms of local data from a triangulation. Potential physical applications of our result exist in light of the fact that the Crane-Yetter invariant is a rigorous version of ideas of Ooguri on B wedge F theory.
hep-th↗