La búsqueda, por parte del ser humano, de bellos diseños, la selección de formas y colores de distintas piezas para sus muros y embaldosados, y la repetición sistemática de motivos produjeron patrones simétricos como ejemplos de teselados. Así mismo, la naturaleza ha encontrado bellísimos teselados resolviendo sus propios problemas. Un teselado embaldosado de una superficie es cubrirla con una misma pieza que se repite sin dejar espacios ni solapamientos. Aunque a simple vista se piense que son infinitas las formas de producir diseños simétricos planos, básicamente existen solo 17 formas de producirlos. Mostraremos que la ejecución de estos teselados sigue unas reglas sencillas y precisas, las cuales hemos utilizado para imaginar 17artefactos, los cuales son ejemplos del concepto debido a William Thurston, deorbifold (orbificie o calidoscopio generalizado) y que pueden ser utilizados en la impresión de cualquier diseño simétrico plano. Exhibiremos estos artefactos por medio de algunos dibujos y utilizaremos algunas de las obras de Escher para ilustrar nuestra conferencia. Se verá que los conceptos de translación, rotación y de reflexión pueden enseñarse fácilmente por medio de la utilización de estos artefactos.

Using a new way to represent links, that we call a buttery repre- sentation, we assign to each 3-bridge link diagram a sequence of six integers, collected as a triple (p=n;q=m;s=l), such that p q s 2, 0 < n p, 0 < m q and 0 < l s. For each 3-bridge link there exists an innite num- ber of 3-bridge diagrams, so we dene an order in the set ( p=n;q=m;s=l) and assign to each 3-bridge link L the minimum among all the triples that corre- spond to a 3-buttery of L, and call it the buttery presentation of L. This presentation extends, in a natural way, the well known Schubert classication of 2-bridge links. We obtain necessary and sucient conditions for a triple ( p=n;q=m;s=l) to correspond to a 3-buttery and so, to a 3-bridge link diagram. Given a triple (p=n;q=m;s=l) we give an algorithm to draw a canonical 3-bridge diagram of the associated link. We present formulas for a 3-buttery of the mirror image of a link, for the connected sum of two rational knots and for some important families of 3-bridge links. We present the open question: When do the triples (p=n;q=m;s=l) and (p 0 =n 0 ;q 0 =m 0 ;s 0 =l 0 ) represent the same 3-bridge link?

With the idea of an eventual classification of 3-bridge links, we define a very nice class of 3-balls (called butterflies) with faces identified by pairs, such that the identification space is S^3, and the image of a prefered set of edges is a link. Several examples are given. We prove that every link can be represented in this way (butterfly representation). We define the butterfly number of a link, and we show that the butterfly number and the bridge number of a link coincide. This is done by defining a move on the butterfly diagram. We give an example of two different butterflies with minimal butterfly number representing the knot 8_20. This raises the problem of finding a set of moves on a butterfly diagram connecting diagrams representing the same link. This is left as an open problem.

Using a new way to represent links, that we call a butterfly representation , we assign to each 3-bridge link diagram a sequence of six integers, collected as a triple (p/n,q/m,s/l) , such that p≥ q≥ s≥2 , 0

Representations of two bridge knot groups in the isometry group of some complete Riemannian 3-manifolds as E(3) (Euclidean 3-space), H(3) (hyperbolic 3-space) and E(2,1) (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 variety of affine c-representations of G. Each point in this variety corresponds to a representation in the unit group of a quaternion algebra and their affine deformations.

It is well known that there are 17 crystallographic groups that determine the possible tessellations of the Euclidean plane. We approach them from an unusual point of view. Corresponding to each crystallographic group there is an orbifold. We show how to think of the orbifolds as artifacts that serve to create tessellations.

An orientable 3-orbifold is universal iff every closed, orientable 3-manifold is the underlying space of an orbifold structure that is an orbifold-covering of it. The first known example of universal orbifold was B-4,B-4,B-4 = (S-3, B, 4) where B denotes the Borromean rings and all the isotropy groups are cyclic of order 4. The main result in this article is that the hyperbolic orbifold B-m,B-2p,B-2q is universal for every m >= 3, p >= 2, q >= 2.

A 2-universal knot is a universal knot having all branching indices 1 or 2. Here we construct the first example of a 2-universal knot which could be a hyperbolic knot (SnapPea). Hence, we could have an example of a universal group generated by 180° rotations.

The complete classification of representations of the Trefoil knot group G in S^{3} and SL(2,R), their affine deformations, and some geometric interpretations of the results, are given. Among other results, we also obtain the classification up to conjugacy of the non cyclic groups of affine Euclidean isometries generated by two isometries $\mu$ and $\nu$ such that $\mu^{2}=\nu^{3}=1$, in particular those which are crystallographic. We also prove that there are no affine crystallographic groups in the three dimensional Minkowski space which are quotients of G.

It has been shown [H. M. Hilden et al., Invent. Math. 87 (1987), no. 3, 441–456;] that the orbifold group U of the Borromean rings with singular angle 90 degrees is universal, i.e. for every closed orientable 3-manifold M3 there is a finite index subgroup G of U such that M3=H3/G. Since the fundamental group of M3 is the quotient of G modulo the subgroup generated by rotations, one would like to classify the finite index subgroups of U. In this paper, the authors begin the classification of the finite index subgroups that are generated by rotations. The group U acts as a group of isometries of hyperbolic 3-space H3 so that there is a tessellation of H3 by regular dodecahedra any one of which is a fundamental domain for U. The authors construct a closely related Euclidean crystallographic group Uˆ corresponding to a tessellation of E3 by cubes that are fundamental domains for Uˆ, and exhibit a homomorphism φ:U→Uˆ which defines a branched covering H3→E3 that respects the two tessellations. They classify the finite index subgroups of Uˆ, and use their pullback under φ to obtain the main result of the paper: For any positive integer n there is an index n subgroup of U generated by rotations.

We give a constructive proof of a Theorem of Izmestiev and Joswig. Namely, given (L, omega) where L is a link in S' and w a simple (not necessarily transitive) representation of pi(1) (S-3\L) onto the symmetric group Sigma(4) of four elements {1, 2, 3,4} we construct a triangulation of S-3 giving rise to (L, omega) in a natural way.

Arithmetic and non arithmetic hyperbolic 3-manifolds having infinitely many fibrations over S-1 of arbitrary high genus are constructed. A concrete example is studied in detail.

If K is a hyperbolic knot in S3, an algebraic component of its character variety containing one holonomy of the complete hyperbolic structure of finite volume of S3∖K is an algebraic curve K. The traces of the peripheral elements of K define polynomial functions in K, which are related in pairs by polynomials (peripheral polynomials). These are determined by just two adjacent peripheral polynomials. The curves defined by the peripheral polynomials are all birationally equivalent to K, with only one possible exception. The canonical peripheral polynomial relating the trace of the meridian with the trace of the canonical longitude of K, is a factor of the A-polynomial.

A triangulation $\Delta$ of $S^{3}$ defines uniquely a number $m\leq4,$ a subgraph $\Gamma$ of $\Delta$ and a representation $\omega(\Delta)$ of $\pi_{1}(S^{3}\backslash\Gamma)$ into $\Sigma_{m.}$ It is shown that every $(K,\omega)$, where $K$ is a knot or link in $S^{3}$ and $\omega$ is transitive representation of $\pi_{1}(S^{3}\backslash K)$ in $\Sigma_{m},$ $2\leq m\leq3,$ equals $\omega(\Delta)$, for some $\Delta$. From this, a representation of closed, orientable 3-manifolds by triangulations of $S^{3}$ is obtained. This is a theorem of Izmestiev and Joswig, but, in contrast with their proof, the methods in this paper are constructive. Some generalizations are given. The method involves a new representation of knots and links, which is called a butterfly representation.

A butterfly is a 3-ball B with an even number of polygonal faces, named wings, pair-wise identified. Each identification between two wings is required to be a topological reflexion whose axis is an edge shared by the wings. The set of axes of the identifications is called the thorax of the butterfly. A knot K⊂S3 admits a butterfly representation if there is a butterfly B with thorax T such that, after the identifications, (B,T) is homeomorphic to (S3,K). In this paper it is shown that any 3-colorable knot admits a butterfly representation (B,T) such that the butterfly B has a 4-colored triangulation compatible with the 3-coloration of the knot. By a result of H. M. Hilden [Amer. J. Math. 98 (1976), no. 4, 989–997;] and J. M. Montesinos [Quart. J. Math. Oxford Ser. (2) 27 (1976), no. 105, 85–94;], one can associate to any 3-manifold a 3-colored knot. A corollary of the main result of the paper is therefore that one can associate to any 3-manifold at least one butterfly.