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How Big is the Kauffman Bracket ?
by
Charles Frohman
Department of Mathematics, The University of Iowa
Jozef Przytycki introduced skein modules as a way of constructing a Jones polynomial for links in an arbitrary three manifold. In this lecture we achieve his goal for links in circle bundles over a surface. Amazingly the polynomial is no longer a polynomial, but a holomorphic function on the unit disk. We also see that our answer extends Turaev's shadow world evaluations at roots of unity.
Here I will recall the definition of the Kauffman bracket, with variable t that will be a complex number. Throughout the lecture we will let s=t2, and q=t4=s2. Finally, we let h be a number with eh=q. We let [n]=[(sn-s-n)/(s-s-1)].
Date received: November 23, 2000
Hyperbolic structure on "link" complements in the 4-sphere
by
Dubravko Ivansic
The George Washington University
There are many known examples of link complements in the 3-sphere that carry a hyperbolic structure, but in one dimension higher none have yet been displayed. For this purpose, the proper generalization of a ``link'' in dimension 4 is a disjoint union of tori and Klein bottles inside a closed 4-manifold.
We prove that some of the examples of hyperbolic 4-manifolds constructed by Ratcliffe and Tschantz have orientable double covers that are complements of tori inside the 4-sphere. Furthermore, we show that they have higher-order covers that are complements of tori inside other simply connected 4-manifolds.
Date received: December 6, 2000
Newton, Klein, Kauffman, Cayley and the Four Color Problem
by
Paul C. Kainen
Department of Mathematics, Georgetown University
An argument is given to show that the problem of map coloring in the plane, well-known to be equivalent to a problem involving pairs of rooted, cubic, plane trees, may be regarded as a generalization of Newtonian mechanics. Coloring is equivalent to the propagation of torque. The latter problem was treated by Felix Klein in his book analyzing the motion of a top. Klein introduces four complex parameters to describe a model. On the other hand, Kauffman noted that if the inputs are regarded as quaternions, rather than axial vectors, then associativity prevents any discrepancy. We have pointed out that the relationship between trees can be considered asymmetrically, with one tree regarded as generating the set of admissable colorings via states (clockwise or counterclockwise) at each vertex. By the now-standard technique of complexification, replace the state at a vertex by an element in the two-dimensional vector space over the complex numbers. Normalizing, the state is now a qubit. Thus, a gedanken quantum computer experiment would always register a solution for any suitable pair of such trees. A complete theory of quantum mechanics must, therefore, contain a solution to the Four Color Problem. We propose an approach based on the unification of geometry and arithmetic proposed by Klein and actually embodied in the octonions. Since it was Cayley who first announced the Four Color Problem to the mathematical community, it is ironic that the ``numbers'' he invented (with Graves) may be required for a natural solution.
Date received: December 1, 2000
A Quantum Obstruction to Embedding
by
Joanna Kania-Bartoszynska
Boise State University
Coauthors: Charlie Frohman
The Jones polynomial of a link was introduced in the mid 1980's. It was defined as the normalized trace of an element of a braid group, corresponding to the link, in a certain representation. Shortly after its introduction, the need for a cut and paste theory to explain the Jones polynomial became understood. Witten realized such a theory using ideas from quantum field theory. His construction of topological quantum field theory rested on deep physical intuitions. In addition to explaining the Jones polynomial, Witten's theory produced a whole new realm of three-manifold invariants.
Although the study of 3-manifold invariants is substantial in itself, few applications to classical 3-manifold topology have been found. We use quantum invariants to develop obstructions to embedding one 3-manifold in another. I will describe these obstruction and illustrate their use with examples.
Date received: December 1, 2000
Shortest vertical geodesics in surgeries on one cusp of the Borromean rings
by
Barbara E. Nimershiem
Franklin & Marshall College
Thurston showed that almost all knots and links in S3 have a complete hyperbolic structure on their complements. An even more surprising result (also by Thurston) is that all but finitely-many Dehn surgeries on a hyperbolic knot or link have a complete hyperbolic structure. This talk will examine, in a very explicit way, these statements for the Borromean rings. Specifically, I will present the standard description of the complete hyperbolic structure on the complement of the Borromean rings. I will also describe an explicit parameterization of the complete structures on the hyperbolic 3-manifolds resulting from Dehn surgeries on one component of the Borromean rings. This parameterization allows us to determine explicitly the shortest geodesic(s) connecting one of the remaining ends to itself. Knowledge of such geodesics enables us to obtain interesting results about lengths of curves in the boundary of a maximal cusp.
Date received: December 5, 2000
Tilings associated with endomorphisms of free groups
by
E. Arthur (Robbie) Robinson
George Washington University
Coauthors: Shunji Ito (Tsuda College), Maki Furukado (Tsuda College)
A tiling x of R2 is called a self-affine if there is an expanding linear mapping B on R2 such that each tile in x is a decomposition of tiles in Bx. Let A be an d×d nonsingular hyperbolic integer matrix, let W denote the expanding subspace of A and let B = A|W. Our goal is to construct a self-affine tiling of W using the method of Dekking.
Let w be an endomorphism of F<d> such that A is the abelianization of w. We view w as a substitution. Note that many substitutions correspond to each A. Let pi, i=1, ..., d, denote the projections of the standard basis in Rd to W. For i ≠ j, B-n(wn [pi, pj]) is a closed curve which converges to a compact set in the Hausdorff topology. The goal is to choose \theta so that B-n(wn[pi, pj]) is a simple closed curve for each n. When this happens, the method of Dekking yields a Jordan curve and by standard methods in the theory of tilings, we can construct a self-affine tiling corresponding to B.
We concentrate on the case where A is a 4×4 matrix satisfying the non-Pisot condition (two eigenvalues on either side of the unit circle). In this case we compute a 6×6 matrix A*' and if A* ≥ 0 we have an algorithm to find a solution. The case A*\not ≥ 0 remains mysterious.
Date received: December 4, 2000
From quantum gravity to Fermat's Last Theorem
by
Adam S. Sikora
CRM/ISM (Montreal)
We will follow a path connecting the theory of spin networks with the work of Wiles and others on pseudo-representations (related to Fermat's Last Theorem). On our way, we will encounter moduli spaces of representations (character varieties), skein modules, Jones polynomial, and quantum invariants of 3-manifolds.
Date received: November 22, 2000
Tangles and persistent dynamical invariants
by
Dan Silver
Univ. of South Alabama and UMD
A tangle t embeds in a link l if some diagram of t extends to a diagram of l. In his Ph.D. thesis D. Krebes proved that if t is a 4-tangle embedded in l such that both the numerator and denominator closures admit nontrivial Fox n-colorings, then l also admits a nontrivial n-coloring. His result was later generalized by D. Silver and S. Williams and also by D. Ruberman. A dynamical system can be associated to a link in a variety of ways. In one such dynamical system Fox n-colorings appear as period-2 points. We discuss symbolic dynamical systems associated to a tangle t that persist in l whenever the tangle embeds. We show that F. Ladrappier's "3-dot example" arises in this way.
Date received: November 29, 2000
p-adic Entropy and Colorings
by
Susan Williams
University of South Alabama
Coauthors: Daniel Silver (University of South Alabama)
New techniques due to G. Everest, D. Lind and T. Ward that combine symbolic dynamics and number theory yield results about branched cover homology of knots.
Date received: December 1, 2000
Symmetry of links and classification of lens spaces
by
Akira Yasuhara
Tokyo Gakugei University and George Washington University
Coauthors: Jozef H. Przytycki (George Washington University)
We give a concise proof of a classification of lens spaces up to orientation-preserving homeomorphisms. Our method is motivated by that of Fox-Brody. While the chief ingredient in their proof was a study of the Alexander polynomial of knots in lens spaces, we study the Alexander polynomial of symmetric' links in S3.
Date received: November 22, 2000
Copyright © 2000 by the author(s). The author(s) of this work and the organizers of the conference have granted their consent to include this abstract in Topology Atlas.