The Full Pythagorean Theorem
We motivate and then prove a generalized pythagorean theorem for parallelepipeds in Euclidean space.
math.HO↗
arXiv subjects
Charles Frohman
Publications and source records attributed to Charles Frohman.
A Note on the Bar-Natan Skein Module
We introduce a new skein module for three manifolds based on properly embedded surfaces and their relations introduced by D.Bar-Natan, and modified by M.Khovanov. We compute the structure of the modules for some manifolds, including Seifert fibred manifolds.
math.QA↗
Shadow world evaluation of the Yang-Mills measure
A new state-sum formula for the evaluation of the Yang-Mills measure in the Kauffman bracket skein algebra of a closed surface is derived. The formula extends the Kauffman bracket to diagrams that lie in surfaces other than the plane. It also extends Turaev's shadow world invariant of links in a circle bundle over a surface away from roots of unity. The limiting behavior of the Yang-Mills measure when the complex parameter approaches -1 is studied. The formula is applied to compute integrals of simple closed curves over the character variety of the surface against Goldman's symplectic measure.
math.GT↗
The Quantum Content of the Normal Surfaces in a Three-Manifold
The formula for the Turaev-Viro invariant of a 3-manifold depends on a complex parameter t. When t is not a root of unity, the formula becomes an infinite sum. This paper analyzes convergence of this sum when t does not lie on the unit circle, in the presence of an efficient triangulation of the three-manifold. The terms of the sum can be indexed by surfaces lying in the three-manifold. The contribution of a surface is largest when the surface is normal and when its genus is the lowest.
math.GT↗
The Topological Classification of Minimal Surfaces in R^3
We give a complete topological classification of minimal surfaces in Euclidian three-space.
math.DG↗
The Yang-Mills Measure in the Kauffman Bracket Skein Module
For each closed, orientable surface F, we construct a local, diffeomorphism invariant trace on the Kauffman bracket skein module K_t(F x [0,1]). The trace is defined when |t| is neither 0 nor 1, and at certain roots of unity. At t = - 1, the trace is integration against the symplectic measure on the SU(2) character variety of the fundamental group of F.
math.GT↗
Quantum Obstruction Theory
We use topological quantum field theory to derive an invariant of a three-manifold with boundary. We then show how to use this invariant as an obstruction to embedding one three-manifold in another.
math.GT↗
The A-polynomial from the noncommutative viewpoint
We develop an invariant of knots that depends on a complex parameter t, describing a left ideal in the noncommutative torus. When the parameter is set equal to -1 we recover the A-polynomial of the knot. We relate the invariant to the colored Jones polynomials of the knot.
math.QA↗
Skein Modules and the Noncommutative Torus
We show that the Kauffman bracket skein module of a cylinder over the torus embeds as a subalgebra of the noncommutative torus. Using this we derive nice formulas for the Jones-Wenzl idempotents and analyze the structure of the Kauffman bracket skein module of the unknot as a module over the Kauffman bracket skein module of a cylinder over the torus.
math.QA↗
Skein Modules and Lattice Gauge Field Theory
See q-alg/9710003 for the corrected version of this paper.
math.GT↗
Topological Interpretations of Lattice Gauge Field Theory
We construct lattice gauge field theory based on a quantum group on a lattice of dimension 1. Innovations include a coalgebra structure on the connections, and an investigation of connections that are not distinguishable by observables. We prove that when the quantum group is a deformation of a connected algebraic group (over the complex numbers), then the algebra of observables forms a deformation quantization of the ring of characters of the fundamental group of the lattice with respect to the corresponding algebraic group. Finally, we investigate lattice gauge field theory based on quantum SL(2,C), and conclude that the algebra of observables is the Kauffman bracket skein module of a cylinder over a surface associated to the lattice.
q-alg↗
Skein Homology
For each skein module we describe a homology theory which, for any three manifold recovers the skein module at its zero level. The theory measures skein-like relations among skein relations, mimicking Hilbert's theory of syzygies. We work explicit examples for the homology groups corresponding to the Kauffman bracket. It is shown, in particular, that for every manifold most of the groups are non-trivial.
q-alg↗
Skein Quantization and Lattice Gauge Field Theory
This is a survey article describing the various ways in which the Kauffman bracket skein module is a quantization of surface group characters. These include a purely heuristic sense of deformation of a presentation, a Poisson quantization, and a lattice gauge field theory quantization.
q-alg↗
Understanding the Kauffman bracket skein module
The Kauffman bracket skein module $K(M)$ of a 3-manifold $M$ is defined over formal power series in the variable $h$ by letting $A=e^{h/4}$. For a compact oriented surface $F$, it is shown that $K(F \times I)$ is a quantization of the $\g$-characters of the fundamental group of $F$, corresponding to a geometrically defined Poisson bracket. Finite type invariants for unoriented knots and links are defined. Topologically free Kauffman bracket modules are shown to generate finite type invariants. It is shown for compact $M$ that $K(M)$ can be generated as a module by cables on a finite set of knots. Moreover, if $M$ contains no incompressible surfaces, the module is finitely generated.
q-alg↗