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A link in S3 is called a universal link if every closed orientable 3-manifold is a branched cover of S3 over this link. It is well known that the Borromean rings and many other links are universal links. The question whether a link is universal can be naturally extended to orbifolds. An orbifold M is said to be universal if every closed orientable 3-manifold is the underlying space of an orbifold which is an orbifold covering of M. Let Bm,n,p denote the orbifold whose underlying space is S3, whose singular set is the Borromean rings B, and whose isotropy groups for the three components of B are cyclic groups of orders m, n and p. In an earlier paper of H. M. Hilden et al. [Invent. Math. 87 (1987), no. 3, 441–456;], it was shown that B4,4,4 is universal. In this paper, the authors generalize this result and prove that Bm,2p,2q is universal for every m≥3, p≥2, q≥2., MTM, Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

Throughout his paper, the author uses "orientable manifold'' to mean a compact connected orientable 3-manifold without boundary. Such a manifold is known to be a ramified covering over a link of the 3-sphere, in which the ramification index of each singular point is ≤2. If the covering has n leaves, suppose that there are m points of index 2 and 2m points of index 1; such a covering is of type (m,n−2m). The author's main theorem states: Every orientable manifold is a ramified covering of type (1,n−2). He also uses the notion of a "link with a colouring of type (m,n−2m)''; these are intimately related to ramified coverings of type (m,n−2m). He conjectures that every link having a colouring of type (1,n−2) is "separable'', a term too complicated to define here. With this conjecture and his main theorem, he enunciates two further theorems and a second conjecture to show that his two conjectures, if true, would imply the Poincaré hypothesis for 3-manifolds. The author adds a note in proof to say that his first conjecture is false, as will be shown in a forthcoming paper by R. H. Fox. It therefore seems unnecessary to detail the conjectures in this review., Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

W. Thurston proved the existence of universal links L⊂S3 which are defined by the property that every closed orientable 3-manifold is a branched covering over L⊂S3. The authors answered earlier [Bull. Amer. Math. Soc. (N.S.) 8 (1983), no. 3, 449–450;] Thurston's question of whether there are universal knots in the affirmative. In the paper under review, they start from the fact that every closed orientable 3-manifold is an irregular 3-fold covering over a negative closed braid, and proceed by changing the braid by certain moves which do not alter the covering manifold. Thus they arrive at the conclusion that the Whitehead link, the Borromean rings and the knot 946 are universal. Whether the figure-eight knot is universal remains an open question., Comisión Asesora de Investigación científica y técnica, Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

General branched coverings, folded coverings, and branched folded coverings are all special cases of the spreads introduced by R. H. Fox in the 1950's [in A symposium in honor of S. Lefschetz, 243–257, Princeton Univ. Press, Princeton, N.J., 1957;] to put the point set topology of these objects on a solid footing. This largely expository article revisits the subject, giving it a full treatment. Certain definitions are extended and several essentially new sufficient conditions for a map to be a spread are given. The concept of a singular covering is introduced, allowing a common treatment of branched coverings and branched folded coverings. One motivation for re-examining and extending the theory is for applications to possibly wild knots. In addition, the theory contains, for example, the end compactification of Freudenthal as a special case, which is given a complete treatment here. The paper opens with a useful, detailed historical survey putting all the work in the area in a common framework., Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

This paper deals with Heegaard splittings and Heegaard diagrams (denoted H-diagrams). Two interesting examples are given which shed light on certain questions about "minimality'' of H-diagrams. An H-diagram is a quadruple (M,F,v,w), where M is a closed orientable 3-manifold, F is a surface embedded in M that separates it into two handlebodies V and W, and v and w are complete systems of meridian discs for V and W. The complexity of the H-diagram, c(M,F,v,w), is the cardinality of ∂v∩∂w. An H-diagram (M,F,v,w) is pseudominimal if c(M,F,v,w)≤c(M,F,v,w′) for all w′ and c(M,F,v,w)≤c(M,F,v′,w) for all v′. It is minimal if c(M,F,v,w)≤c(M,F,v′,w′) for all v′ and w′. In the first example, two H-diagrams of the lens space M=L(7,2) are given with different complexity. This shows that pseudominimality does not imply minimality. In this example, the H-diagram has a pair of cancelling handles. F. Waldhausen asked the question: "In an H-diagram which is pseudominimal but not minimal is there always a pair of cancelling handles?'' The second example shows that either (a) there is a 3-manifold with two minimal H-splittings of different genus or (b) there is an H-diagram that is pseudominimal but not minimal and has no pair of cancelling handles. The authors conjecture that (a) holds. This is an enlightening paper to read for anyone wishing to learn some of the methods and techniques of Heegaard splittings., National Science Foundation, Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

An n-knot (Sn+2,Sn) is said to be amphicheiral if there is an orientation-reversing autohomeomorphism of Sn+2 which leaves Sn invariant as a set. An n-knot is said to be invertible if there is an orientation-preserving autohomeomorphism of Sn+2 whose restriction to Sn is an orientation-reversing autohomeomorphism of Sn. The authors prove that for any integer n there are smooth n-knots which are neither amphicheiral nor invertible. Actually, they prove it for n≥2, referring to the paper of H. F. Trotter [Topology 2 (1963), 275–280; errata, MR 30, p. 1205] for the case n=1. The methods employed are mainly algebraic, involving for example the duality pairings of R. C. Blanchfield and J. Levine, and in most cases the work of previous authors is used to guarantee the existence of knots with the desired algebraic properties., Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

In an unpublished preprint W. Thurston showed the existence of a six component link in the 3-sphere such that every three-manifold can be expressed as a branched cover of the 3-sphere branched over this link. He called links with this property "universal links'' and asked if a universal knot exists. The authors confirm this by showing that any link can be embedded in the branch locus of a 3-sphere branched covering of a 3-sphere branched over a knot, Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

In 1962, R. H. Fox asked [Topology of 3-manifolds and related topics (Proc. Univ. Georgia Inst., 1961), pp. 168–176, especially pp. 175–176, Prentice-Hall, Englewood Cliffs, N.J., 1962)] whether a 2-knot group could have infinitely many ends. The authors answer this question in the affirmative by exhibiting 2-knots whose groups have infinitely many ends., Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

Representations of two bridge knot groups in the isometry group of some complete Riemannian 3-manifolds as E3 (Euclidean 3-space), H3 (hyperbolic 3-space) and E2, 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 ofG. Each point in this variety corresponds to a representation in the unit group of a quaternion algebra and their affine deformations., MTM2007-67908-C02-01 and MTM2010-21740-C02-02., MTM2006-00825 and MTM2009-07030., Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

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., COLCIENCIAS, DIME, Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

A link or knot in S 3 is universal if it serves as common branching set for all closed, oriented 3-manifolds. A knot is simple if its exterior space is simple, i.e. any incompressible torus or annulus is parallel to the boundary. No iterated torus knot or link is universal, but we know of many knots and links that are universal. The natural problem is to describe the class of universal knots, and this was asked by one of the authors in his address to the `Symposium of Kleinian groups, 3-manifolds and Hyperbolic Geometry' held in Durham, U. K., during July 1984. In the problem session of the same symposium W. Thurston asked if a non-simple knot can be universal and more concretely, if a cable knot can be universal. The question had the interest of testing whether the universality property has anything to do with the hyperbolic structure of some knots. That this is not the case is shown in this paper, where we give infinitely many examples of double, composite and cable knots that are universal., N.S.F., Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

J. S. Birman [same J. 28 (1976), no. 2, 264–290] has shown that any two plat representations of a link in S3 are stably equivalent and that stabilization is a necessary feature of the equivalence for certain composite knots. She has asked whether all 2n-plat representations of a prime link are equivalent. The author provides a negative answer, by exhibiting an infinite collection of prime knots and links in S3 in which each element L has at least two minimal and inequivalent 6-plat representations. In addition, as an application of another result of Birman [Knots, groups and 3-manifolds (Papers dedicated to the memory of R. H. Fox), pp. 137–164, Ann. of Math. Studies, No. 84, Princeton Univ. Press, Princeton, N.J., 1975], the 2-fold cyclic covering spaces of S3 branched over such links L form further examples of closed, orientable, prime 3-manifolds having inequivalent minimal Heegaard splittings, which were first constructed by Birman, F. González-Acuña and the author [Michigan Math. J. 23 (1976), no. 2, 97–103]., Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

Dedicated with respect and friendship to Professor Yukio Matsumoto on his 60th birthday, In this paper, the possibility of embedding a nontrivial string (R3,K) in the trivial knot (S3,U) is investigated. Uncountably many examples are given. The complementary space in S3 of the image of R3 under the embedding is a continuum. Some well-known snake-like continua appear as these residual spaces. The 2-fold coverings of R3 branched over the strings involved are studied. As a consequence, concrete descriptions of the p-adic solenoids are given, and it is shown that the Whitehead continuum is homeomorphic to Bing's snake-like continuum without end points., Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

The authors prove that every geometric orbifold is good. More precisely, let X be a smooth connected manifold, and let G be a group of diffeomorphisms of X with the property that if any two elements of G agree on a nonempty open subset of X, then they coincide on X. If Q is an orbifold which is locally modelled on quotients of open subsets of X by finite subgroups of G, then the authors prove that the universal orbifold covering of Q is a (G,X)-manifold. A similar theorem was stated, and the proof sketched, in W. Thurston's lecture notes on the geometry and topology of 3-manifolds., Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

Forma parte de la "Décima edición del Programa de Promoción de la Cultura Científica y Tecnológica, organizado por la Real Academia de Ciencias Exactas, Físicas y Naturales", La cristalografía es una disciplina de la geología que estudia los cristales: sus formas externas e internas; las posibilidades de estas formas, etc. Los aspectos matemáticos de la teoría son de extraordinario interés para los matemáticos porque admiten generalizaciones en varias direcciones. En primer lugar, pueden estudiarse los grupos cristalográficos n dimensionales para todo n>0. Algunos grupos de dimensiones altas (5 ó más) han aclarado la estructura de algunos cuasicristales. Pero la generalización más interesante consiste en el estudio de los grupos cristalográficos en espacios no euclidianos. El estudio de ellos, en el plano hiperbólico, por ejemplo, tiene relaciones muy importantes (que se remontan a Gauss) con la teoría de números. El caso tridimensional está relacionado con la teoría de nudos, etc. En esta conferencia introduciré estas ideas empleando útiles topológicos e ilustrándolos mediante fotografías de flores, insectos y obras de arte., Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

The author shows that every compact connected oriented 3-manifold, after capping off boundary components by cones, is a covering of S3 branched over the 1-complex G which is "a pair of eyeglasses''. The author gives algorithms for passing between a Heegaard decomposition of a 3-manifold and this covering description. He also determines necessary and sufficient conditions for such a covering to have cone singularities. In a paper by W. Thurston ["Universal links'', Preprint], a link with similar properties (for closed 3-manifolds) to G is constructed., Comisión Asesora del Ministerio de Educación y Ciencia, Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

Dedicado a María Teresa Lozano Imízcoz tras 27 años de fructífera colaboración, This is a survey article discussing the author's work in a series of several publications on the relationship between 3-manifolds and wild knots in the 3-sphere and strings in R3 given by branched coverings. He includes an introduction to ordinary combinatorial branched coverings and a parallel introduction to the general topological branched coverings defined by R. H. Fox. Among other things one may conclude that every closed, oriented 3-manifold is a 3-fold covering of the 3-sphere branched over a wild knot. The paper ends with a brief discussion of two open problems., Colciencias, MTM, Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

A smooth n-knot K in Sn+2 is said to be quasiaspherical if Hn+1(U)=0, where U is the universal cover of the exterior of K. Let G be the group of K and H the subgroup generated by a meridian. Then (G,H) is said to be unsplittable if G does not have a free product with amalgamation decomposition A∗FB with F finite and H contained in A. The authors prove that K is quasiaspherical if and only if (G,H) is unsplittable. If the group of K has a finite number of ends, then K is quasiaspherical and it was conjectured by the reviewer [J. Pure Appl. Algebra 20 (1981), no. 3, 317–324; MR0604323 (82j:57019)] that the converse was true. The authors give a very nice and useful method of constructing knots in Sn+2 and apply this method to produce counterexamples to the conjecture., Comission Asesora del MUI, Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

Applied to a knot K of S1 in S3, Zeeman's n-twist-spinning construction produces a knot Kn of S2 in S4 [ E. C. Zeeman , Trans. Amer. Math. Soc. 115 (1965), 471–495;]. Here the author gives an explicit "movie presentation'' of Kn, that is, a description of Kn in terms of the cross-sections Kn∩S3×{t}., Comité Conjunto Hispano-Norteamericano, NSF, Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

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., Fulbright Colombia, COLCIENCIAS, Universidad Nacional de Colombia, Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

The authors construct a cover S3→S3 branched over the "figure eight" knot with preimage the "roman link" and a cover S3→S3 branched over the roman link with preimage containing the Borromean rings L. Since L is universal (i.e. every closed, orientable 3-manifold can be represented as a covering of S3 branched over L) it follows that the "figure eight'' knot is universal, thereby answering a question of Thurston in the affirmative. More generally, it is shown that every rational knot or link which is not toroidal is universal, Comisión Asesora de Investigación Científica y Técnica., Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

It is proved that any closed orientable 3-manifold is a 3-fold irregular branched covering of the 3-sphere branched over a wildly embedded knot. These branched coverings are obtained by starting with such a branched covering over a tame knot and then inserting into it a particular irregular branched covering of the 3-sphere over the 3-sphere, with a wild branch set. It is also shown how to use related techniques to produce branched coverings of certain open 3-manifolds over tame, properly embedded arcs in R3. For example, the Whitehead contractible open 3-manifold is expressible as a 2-fold branched covering over such an arc, conjecturally in uncountably many different ways. These results should be considered as illustrations of the general construction given in the author's recent paper [Rev. Mat. Complut. 15 (2002), no. 2, 533–542, Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

It is announced that the Freudenthal compactification of an open, connected, oriented 3-manifold is a 3-fold branched covering of S 3 . The branching set is as nice as can be expected. Some applications are given., Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

Jornadas Matemáticas Hispano-Lusitanas (7. 1980. Sant Feliú de Guixols). Part II, From the text: "We deal with coverings of the 3-sphere S3branched over a link (= a system of knots) as a way of representing closed orientable 3-manifolds. A simplicial mapping f:Mn→Nn between two compact triangulated n-manifolds M and N is called a branched covering if it is an ordinary covering outside of the (n−2)-skeleton of Nn. The points of Nn whose preimages have fewer points than the covering has leaves form a subcomplex Bn−2 called the branch locus. We use the phrase `f is a covering of Nn branched over B'. "Because this is an expository article, we deal only with some selected topics that help to give an idea of the theory. The material is generally known, although some results are new.'', Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

It is a celebrated result of H. Hilden and the author of the present paper that every closed, connected, oriented 3-manifold is a 3-fold irregular (dihedral) branched covering of the 3-sphere, branched over a knot. Here the author explores a generalization of this result to the case of non-compact manifolds. It is shown that a non-compact, connected, oriented 3-manifold is a 3-fold irregular branched covering of an open subspace of S3, branched over a locally finite family of proper arcs. The branched covering is constructed in such a way that it extends to a branched covering (suitably understood) of the Freudenthal end compactification over the entire 3-sphere. In particular all (uncountably many) contractible open 3-manifolds may be expressed as 3-fold branched coverings of R3, branched over a locally finite collection of proper arcs., Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

It is known that every closed, orientable 3-manifold contains a fi-bered knot—a simple closed curve whose complement is a surface bundle over S1. For K such a fibered knot in a rational homology 3-sphere M it is shown that for any compact submanifold X of M containing K as a null-homologous subset, each component of ∂X is compressible in M−K. If K is a doubled knot (bounds a disk with one clasp) then it follows that K is a double of the trivial knot. More generally, it follows that the genus of X (minimum number of one-handles) is less than the genus of M., NSF, Comité Conjunto Hispano-Norteamericano, Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

J. Birman had observed that the homological information about a given Heegaard splitting of genus g is contained in a double coset in the group of symplectic 2g×2g integer matrices with respect to a suitable subgroup, and found a determinant invariant of this double coset. We obtain complete invariants of these double cosets by characterizing it in terms of the linking form of the manifold lifted to a handlebody of the Heegaard splitting and then finding complete invariants of this lifted form., Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

The Chern-Simons invariant was extended to 3-dimensional geometric cone manifolds in [H. M. Hilden, M. T. Lozano and J. M. Montesinos-Amilibia, J. Math. Sci. Univ. Tokyo 3 (1996), no. 3, 723–744; MR1432115 (98h:57056)]. The present paper is about the behavior of this generalized invariant under change of orientation and with respect to virtually regular coverings. (A virtually regular cover is a cover with the property that the branching index is constant along the fiber over each point of the branching set.) As one might suspect, CS(−M)=−CS(M). However, unlike the volume, the Chern-Simons invariant is not multiplicative with respect to branched coverings. There is a correction term depending on the intersection number of longitudes of inverse images of the singular set with the inverse image of the longitude of the singular set. The paper concludes with applications of the main formula to specific examples., Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

Let M denote a p-fold, branched, cyclic, covering space of S3, and suppose that the three-dimensional Smith conjecture is true for p-periodic autohomeomorphisms of S3. J. S. Birman and H. M. Hilden have constructed an algorithm for deciding whether M is homeomorphic to S3 [Bull. Amer. Math. Soc. 79 (1973), 1006–1010]. Now every closed, orientable three-manifold is a three-fold covering space of S3 branched over a knot [Hilden, ibid. 80 (1974), 1243–1244], but, in the present paper, the author shows that, if Fg is a closed, orientable surface of genus g≥1, then Fg×S1 is not a p-fold, branched cyclic covering space of S3 for any p. As the author points out, this was previously known for p=2 [R. H. Fox, Mat. Hisp.-Amer. (4) 32 (1972), 158–166; the author, Bol. Soc. Mat. Mexicana (2) 18 (1973), 1–32]., Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

Dedicado a Francisco González Acuña en su sexagésimo cumpleaños, According to the confirmed Smith Conjecture [The Smith conjecture (New York, 1979), Academic Press, Orlando, FL, 1984;], a tame knot in the 3-sphere has a cyclic branched covering that is also the 3-sphere only if it is trivial. Here the author produces a nontrivial, wild knot whose n-fold cyclic branched cover is S3, for all n. In fact there are uncountably many inequivalent knots with this property, and the knots can be chosen to bound an embedded disk that is tame in its interior. One might conjecture that any wild knot whose nontrivial n-fold cyclic branched cover is S3 must bound such a disk that is tame in its interior., Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

We give a different proof of the result of M. Sakuma [Math. Sem. Notes Kobe Univ. 9 (1981), no. 1, 159–180] that every closed, oriented 3-manifold M has a 2-fold branched covering space N which is a surface bundle over S1. We also give a new proof of the result of Brooks that N can be made hyperbolic. We give examples of irreducible 3-manifolds which can be represented as 2m-fold cyclic branched coverings of S3 for a number of different m's as big as we like., Comité Conjunto Hispano-Norteamericano, NSF, Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

An important open question about 3-manifolds is whether or not there exists an algorithm for recognizing S3. The author poses two questions about Heegaard diagrams of S3, appropriate answers to either of which would give such an algorithm. If a Heegaard diagram contains either a wave or a cancelling pair, then one can find an equivalent diagram of smaller complexity (in the latter case, of smaller genus). Every nontrivial genus-2 diagram of S3 contains a wave [T. Homma, M. Ochiai and M. Takahashi Osaka J. Math 17 (1980), no. 3, 625–648; MR0591141 (82i:57013)], but this is false for higher genera. The author's first question is whether there are any Heegaard diagrams of S3 without waves and without cancelling pairs. {Reviewer's remark: An example of such a diagram is contained in an article of Ochiai [ibid. 22 (1985), no. 4, 871–873; MR0815455 (87a:57020)].} Given a Heegaard diagram, there is a reduction procedure which produces a so-called pseudominimal diagram. W. Haken [in Topology of manifolds (Athens, Ga., 1969), 140–152, Markham, Chicago, Ill., 1970; MR0273624 (42 #8501)] has suggested that perhaps the only pseudominimal diagrams of S3 are the trivial ones; no counterexamples are known. The author suggests a further reduction step which might be applied to a pseudominimal diagram, yielding several partial diagrams. If any of these has a cancelling pair, then the genus of the original diagram can be reduced. An example is given to show that, in general, for manifolds different from S3, even this enhanced procedure does not always detect the reducibility of a Heegaard splitting. The author's second question, however, is whether it does for splittings of S3. Thus the author is suggesting a possible algorithm for recognizing S3 which allows for the existence of nontrivial pseudominimal diagrams of S3., Comité Conjunto Hispano-Norteamericano, NSF, Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

This is an exposition of the interrelation between orbifolds and crystallographic groups of the plane, focussing especially on patterns that occur in the Alhambra in Granada. This material appears in English in the author's book [Classical tessellations and three-manifolds, Springer, Berlin, 1987;]., Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

W. P. Thurston [Mem. Amer. Math. Soc. 59 (1986), no. 339, i–vi and 99–130;] showed that if a hyperbolic 3-manifold with b1>1 fibers over S1, then it fibers in infinitely many different ways. In this paper, the authors consider a certain family of hyperbolic manifolds, obtained as branched covers of the 3-torus. They show explicitly that each manifold in this family has infinitely many different fibrations., Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

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. Rolfsen's book [Knots and links, Publish or Perish, Berkeley, Calif., 1976, Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

A link L is universal if every closed orientable 3-manifold M is a finite branched covering of S3 with the branch set equal to L. Known examples of universal links are the figure eight knot and the Borromean rings. It is also known that the trefoil knot is not universal. The notion of universality can be refined by allowing only certain types of branching. A branched covering p:M→N over a link L is of type {1,2} if the branching (ramification) index of each component of p−1(K) is either 1 or 2. A link L is 2-universal if every closed orientable 3-manifold M is a finite branched cover of S3 over L, of type {1,2}. The existence of 2-universal links has been known, but not of 2-universal knots. The authors use the existence of 2-universal links to prove that 2-universal knots exist. Their main theorem implies that for any 2-universal link L there exists a branched covering p:S3→S3 over a knot K, such that L is a sublink of the pseudo-branch cover, i.e., such that L is contained in the union of those components of p−1(K) which have branching indices 1. The link L is 2-universal; therefore for any given closed orientable 3-manifold M there exists a branched covering q:M→S3 over L, of type {1,2}. Furthermore, since L is a sublink of the pseudo-branch cover, the branched covering p∘q:M→S3 is also of type {1,2}, implying that K is 2-universal. Part of the reason for studying 2-universal knots is the following. There exists a discrete universal group of hyperbolic isometries U (a discrete group of hyperbolic isometries G is universal if any closed orientable 3-manifold M is homeomorphic to the orbit space H3/H where H is a subgroup of G of finite index). The group U is generated by three 90∘ rotations. Since there are 3-manifolds which are not hyperbolic, any universal group has to contain rotations. The existence of a 2-universal hyperbolic knot would imply the existence of a discrete universal group of hyperbolic isometries that would only contain rotations by 180∘., Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

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., Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

The authors discuss a class of flows on 3-manifolds closely related to Anosov flows, which they call singular Anosov flows. These are flows which are Anosov outside of a finite number of periodic "singular orbits'', such that each singular orbit has a Poincaré section on which the first return map has an "n-pronged singularity'' for some n≥1, n≠2. If only 1-pronged singularities occur the flow is called V-Anosov; the authors observe, for example, that the geodesic flow of a compact, hyperbolic 2-orbifold is V-Anosov. The main theorem is that every closed 3-manifold has a singular Anosov flow. The theorem is proved by constructing a certain link L in the 3-sphere such that L is a universal branching link, so every closed 3-manifold M is a branched cover of the 3-sphere branched over L, and L is the set of singular orbits of some V-Anosov flow on S3, so the lifted flow is a singular Anosov flow on M. In the literature, a singular Anosov flow whose n-pronged singularities always satisfy n≥3 is called pseudo-Anosov. The main theorem should be contrasted with the fact that an Anosov or pseudo-Anosov flow can only occur on an aspherical 3-manifold—an irreducible 3-manifold with infinite fundamental group. The literature contains many constructions of Anosov and pseudo-Anosov flows, but it remains unknown exactly which aspherical 3-manifolds support such flows, Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

A link L of the 3-sphere S3 is said to be g-periodic (g≥2 an integer) if there exists an orientation preserving auto-homeomorphism h of S3 such that h(L)=L, h is of order g and the set of fixed points of h is a circle disjoint from L. A knot is called periodic with rational quotient if it is obtained as the preimage of one component of a 2-bridge link by a g-fold cyclic covering branched on the other component. In this paper the authors introduce a method to compute the excellent component of the character variety of periodic knots (note that for hyperbolic knots the excellent component of the character curve contains the complete hyperbolic structure). Among other examples, this method is applied to the seven hyperbolic periodic knots with rational quotient in Rolfsen's table and with bridge number greater than 2., Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

DGICYT, Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub