María Teresa Lozano

ON THE ARITHMETIC 2-BRIDGE KNOTS AND LINK ORBIFOLDS AND A NEW KNOT INVARIANT

Let (p/q, n) denote the orbifold with singular set the two bridge knot or link p/q and isotropy group cyclic of orden n. An algebraic curve (set of zeroes of a polynomial r(x, z)) is associated to p/q parametrizing the representations of in PSL . The coordinates x, z, are trace(A2)=x, trace(AB)=z where A and B are the images of canonical generators a, b of . Let (xn, zn) be the point of corresponding to the hyperbolic orbifold (p/q, n). We prove the following result: The (orbifold) fundamental group of (p/q, n) is arithmetic if and only if the field Q(xn, zn) has exactly one complex place and ϕ(xn)<ϕ(zn)<2 for every real embedding . Consider the angle α for which the cone-manifold (p/q, α) is euclidean. We prove that 2cosα is an algebraic number. Its minimal polynomial (called the h-polynomial) is then a knot invariant. We indicate how to generalize this h-polynomial invariant for any hyperbolic knot. Finally, we compute h-polynomials and arithmeticity of (p/q, n) with p≦40, and (p/q, n) with p≦99q2≡1 mod p. We finish the paper with some open problems.

VOLUMES AND CHERN-SIMONS INVARIANTS OF CYCLIC COVERINGS OVER RATIONAL KNOTS

In this paper, the authors compute the volumes and Chern-Simons invariants for a class of hyperbolic 3-manifolds, namely, the n-fold branched covers of S3 along the 2-bridge knots p/q. The computation is based on the formula of Schlaffli. In a 1-parameter family of polytopes in a space of constant curvature K, KdV=(1/2)∑lidαi, where V is the volume, and the sum is taken over all edges, li is the length of the ith edge and αi is its dihedral angle. Thus the volume of a 1-parameter family of cone-manifolds can be computed in terms of an initial volume and an integration involving length and cone angle of the singular curves. Similarly, the Chern-Simons invariant can be expressed in terms of an initial value and an integration involving the jump and the angle, based on earlier work of the authors. The 1-parameter family of cone-manifolds arises from the following. It is well-known that these 2-bridge knots have hyperbolic complements, which can be considered as hyperbolic cone-manifold structures on S3 with cone-angle 0 around the knot. It is also well-known that the 2-fold branched cover of S3 along p/q is the lens space Lp,q, which has spherical geometry, which induces a spherical cone-manifold structure on S3 with cone-angle π around the knot. These two structures are members of the family of cone-manifold structures on S3 having the 2-bridge knot p/q as a singular curve with angle α (0≤α≤π). There is an angle αh such that the cone structure is hyperbolic when 0≤α<αh, Euclidean when α=αh, and spherical when αh<α≤π. The authors choose the parameter to be x=2cosα, where α is the cone angle around the knot. They compute the functions jump, β(x), and length, δ(x), from the excellent component of the curve of representations of the knot group into SL(2,C). This allows them to compute the volumes and Chern-Simons invariants of the cone manifolds in terms of explicit integrals. The computation of the covering manifolds follows from the multiplicity of these invariants. Examples of numerical computations are shown at the end.

CHARACTER VARIETIES AND PERIPHERAL POLYNOMIALS OF A CLASS OF KNOTS

The representation space or character variety of a finitely generated group is easy to define but difficult to do explicit computations with. The fundamental group of a knot can have two interesting representations into PSL2(C) coming from oppositely oriented complete hyperbolic structures. These two representations lift to give four excellent SL2(C) representations. The excellent curves of a knot are the components of the SL2(C) character variety containing the excellent representations. It is possible to compute geometric invariants of hyperbolic cone manifolds from suitable descriptions of the excellent curve. In this paper, Hilden, Lozano and Montesinos describe a method for analyzing the character varieties of a large class of knots. The main ingredients in this method are a non-obvious, but convenient parametrization of 2×2 complex matrices and an explicit computation relating the holonomies of the four punctures of a four punctured sphere. In order to qualify when their method will work, Hilden, Lozano and Montesinos introduce the notion of a 2n-net. A 2n-net is an interesting generalization of a 2n-plat. Recall that a 2n-plat is obtained by separately closing the top and the bottom of a 2n-strand braid. A 2n-net is the generalization obtained by allowing rational tangles at the crossing points. The given method to analyze the character variety works for any knot with a 4-net description. The method is remarkably robust. For example, it works for essentially every knot in the table in D. Rolfsens book [Knots and links, Publish or Perish, Berkeley, Calif., 1976

ON REPRESENTATIONS OF 2-BRIDGE KNOT GROUPS IN QUATERNION ALGEBRAS

Representations of two bridge knot groups in the isometry group of some complete Riemannian 3-manifolds as

Geometric conemanifold structures on &#x1d54b;p/q, the result of p/q surgery in the left-handed trefoil knot &#x1d54b;

E^{3}

Harmonic manifolds and embedded surfaces arising from a super regular tesselation

(Euclidean 3-space),

On continuous families of geometric Seifert conemanifold structures

H^{3}

On representations of 2-bridge knot groups in quaternion algebras II: The case of the Trefoil knot group

(hyperbolic 3-space) and

On knots that are universal

E^{2,1}

A Characterization of Arithmetic Subgroups of SL (2, ℝ) and SL (2, ℂ)

(Minkowski 3-space), using quaternion algebra theory, are studied. We study the different representations of a 2-generator group in which the generators are send to conjugate elements, by analyzing the points of an algebraic variety, that we call the \emph{variety of affine c-representations of}