Person: Montesinos Amilibia, José María
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First Name
José María
Last Name
Montesinos Amilibia
Affiliation
Universidad Complutense de Madrid
Faculty / Institute
Ciencias Matemáticas
Department
Area
Geometría y Topología
Identifiers
112 results
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Now showing 1 - 10 of 112
Item On the character variety of group representations of a 2-bridge link p/3 into PSL(2,C)(Boletín de la Sociedad Matemática Mexicana. Segunda Serie, 1992) Hilden, Hugh Michael; Lozano Imízcoz, María Teresa; Montesinos Amilibia, José MaríaConsider the group G of a classical knot or link in S3. It is natural to consider the representations of G into PSL(2,C). The set of conjugacy classes of nonabelian representations is a closed algebraic set called the character variety (of representations of G into PSL(2,C)). If G is the group of a 2-bridge knot or link, then a polynomial results by an earlier published theorem of the authors. This polynomial is related to the Morgan-Voyce polynomials Bn(z), which can be defined by the formulas pn(z)=Bn(z−2), where pn=zpn−1−pn−2, p0=1, p1=z, or (z1−10)n=(pnpn−1−pn−1−pn−2). In this paper the authors do many calculations for classes of 2-bridge knots or links.Item Three manifolds as geometric branched coverings of the three sphere.(Boletín de la Sociedad Matemática Mexicana. Tercera Serie, 2008) Brumfield, G.; Hilden, Hugh Michael; Lozano Imízcoz, María Teresa; Montesinos Amilibia, José María; Ramírez Losada, E.; Short, H.; Tejada Cazorla, Juan Antonio; Toro, M.A finite covolume, discrete group of hyperbolic isometries U, acting on H3, is said to be universal if for every closed orientable 3-manifold M3 there is a finite index subgroup G of U so that M3=H3/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 and that H3/U=S3. In the present paper the authors construct a sequence of hyperbolic orbifold structures on S3 with orbifold groups Gi, i=1,…,4, such that G⊂G1⊂G2⊂G3⊂G4⊂U and they use this to obtain the following geometric branched covering space theorem: Let M3 be a closed orientable 3-manifold. Then there are finite index subgroups G⊂G1 of U such that M3=H3/G, S3=H3/G1 and the inclusion G→G1 induces a 3-fold simple branched covering M3→S3. 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.Item On finite index subgroups of a universal group(Boletín de la Sociedad Matemática Mexicana. Tercera Serie, 2008) Brumfield, G.; Hilden, Hugh Michael; Lozano Imízcoz, María Teresa; Montesinos Amilibia, José María; Ramírez Losada, E.; Short, H.; Tejada Cazorla, Juan Antonio; Toro, M.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.Item Non-Euclidean symmetries of first-order optical systems(Journal of the Optical Society of America, 2020) Monzón Serrano, Juan José; Montesinos Amilibia, José María; Sánchez Soto, Luis LorenzoWe revisit the basic aspects of first-order optical systems from a geometrical viewpoint. In the paraxial regime, there is a wide family of beams for which the action of these systems can be represented as a Möbius transformation. We examine this action from the perspective of non-Euclidean hyperbolic geometry and resort to the isometric-circle method to decompose it as a reflection followed by an inversion in a circle. We elucidate the physical meaning of these geometrical operations for basic elements, such as free propagation and thin lenses, and link them with physical parameters of the system.Item Discrepancy between the rank and the Heegaard genus of a 3-manifold. (Spanish: La discrepancia entre el rango y el genero de Heegaard de una 3-variedad)(Note di Matematica, 1989) Montesinos Amilibia, José MaríaLet M be a closed orientable 3-manifold. The rank of M, rk(M), is the minimum number of elements that can generate π1(M). Clearly rk(M)≤Hg(M), where Hg(M) is the Heegaard genus of M. F. Waldhausen conjectured that equality holds here, but then. Boileau and H. Zieschang [Invent. Math. 76 (1984), no. 3, 455–468;] found an infinite set of (Seifert) manifolds M with Hg(M)=3 and rk(M)=2. To prove the latter equality they started with the presentation of π1(M) resulting from a Heegaard splitting of M of genus 3; they then reduced this presentation to a presentation with only two generators. In the present paper the author shows that one can perform such a reduction by using only Nielsen moves (which are in general not sufficient for transforming an arbitrary finite presentation to any other finite presentation of the same group). Actually he proves a more general theorem about reducing, by Nielsen moves alone, the number of generators in certain presentations coming from Heegaard splittings, and because of this theorem he believes that the strict inequality rk(M)Item Seifert manifolds that are ramified two-sheeted cyclic coverings. (Spanish)(Boletín de la Sociedad Matemática Mexicana. Segunda Serie, 1973) Montesinos Amilibia, José MaríaIf L is a link in the 3-sphere S3, let e:L˜→S3 denote the 2-fold cyclic covering of S3 branched over L. R. H. Fox [Rev. Mat. Hisp.-Amer. (4) 32 (1972), 158–166;] has shown that there is no link L in S3 such that L˜ is S1×S1×S1; the author [ibid. (4) 33 (1973), 32–35] has extended this to Fg×S1 (g≥1), where Fg denotes a closed orientable surface of genus g. In the present article he investigates the following more general question: Given any orientable Seifert fibre space M, determine whether M is homeomorphic to L˜ for some link L⊂S3; if the answer is yes, describe L. He finds an affirmative answer for all orientable Seifert fibre spaces over a 2-sphere or over a nonorientable closed surface as base B. In these cases a corresponding link L is constructed by using the technique of tangle modification introduced by J. H. Conway [Computational problems in abstract algebra (Proc. Conf., Oxford, 1967), pp. 329–358, Pergamon, Oxford, 1970;], to which corresponds the operation of removing from L˜ a solid torus and sewing it back differently in the covering. For orientable base B of positive genus g, i.e., B=Fg (g≥1), the situation is more complex: (i) The author finds a negative answer to the above question for the fibre spaces (Oog|b) without exceptional fibres, provided b≠±1,±2 and g≥1 (for the notation, see H. Seifert's article [Acta Math. 60 (1933), 147–238; Zbl 6, 83]). (ii) Analyzing the special assumption that the unique nontrivial covering transformation of the 2-fold cover is fibre-preserving, the author obtains a list of Seifert fibre spaces with base Fg, each of which is homeomorphic to L˜ for an appropriate link L in S3. (iii) The verification that this list is complete would depend on an affirmative answer to an unsolved question concerning involutions in Seifert fibre spaces. (iv) Modifying the main question, the author proves that each orientable Seifert fibre space over Fg (g≥0) is a 2-fold cyclic cover branched over a link of Hg, the 3-sphere with g handles attached. Finally, it is shown how some of these results extend from the class of Seifert fibre spaces to the class of "graph-manifolds'' introduced by F. Waldhausen [Invent. Math. 3 (1967), 308–333; ibid. 4 (1967), 87–117;]. The paper is a fine piece of geometry, being specified throughout with interesting examples.Item Surgery on double knots and symmetries(Mathematische Annalen, 1987) Montesinos Amilibia, José María; Boileau, Michel; González Acuña, Francisco JavierW. Whitten conjectured [Pacific J. Math. 97 (1981), no. 1, 209–216] that no 3-manifold obtained by a nontrivial surgery on a double of a noninvertible knot is a 2-fold branched covering of S3. The authors give counterexamples to this conjecture and determine the exact range of validity of the conjecture. More generally, they consider closed, orientable 3-manifolds obtained by nontrivial Dehn surgery on a double of a non-strongly invertible knot and study the symmetries of such manifolds, i.e. the homeomorphisms of finite order on these manifolds. They show that, except for a finite number of surgeries, these manifolds admit no (nontrivial) symmetry.Item Non-Euclidean geometries: Gauss, Lobachevskiĭ and Bolyai. (Spanish:Las Geometrías no euclídeas: Gauss, Lobatchewsky y Bolyai)(Historia de la matemática en el siglo XIX. Parte 1, 1992) Montesinos Amilibia, José MaríaThe author presents the origin of non-Euclidean geometry in the works of J. Bolyai, C. F. Gauss and N. I. Lobachevskiĭ. The main ideas from the Appendix by Bolyai (1831) and from Pangeometria by Lobachevskiĭ (posthumous, 1855) and the reaction of Gauss to the publications of Lobachevskiĭ and Bolyai are presented. The paper contains many excerpts from the works of Lobachevskiĭ and Bolyai and from the letters of Gauss and Bolyai. The exposition is illustrated with many figures. Brief biographies of Bolyai and Lobachevskiĭ are also included.Item Around the Borromean link(Revista de la Real Academia de Ciencias Exactas Fisicas y Naturales Serie A: Matemáticas, 2008) Montesinos Amilibia, José MaríaThis is a survey of some consequences of the fact that the fundamental group of the orbifold with singular set the Borromean link and isotropy cyclic of order 4 is a universal Kleinian groupItem On volumes and Chern-Simons invariants of geometric 3-manifolds(Journal of Mathematical Sciences. The University of Tokyo, 1996) Hilden, Hugh Michael; Lozano Imízcoz, María Teresa; Montesinos Amilibia, José MaríaThis paper presents a technique for computing Chern-Simons invariants of certain kinds of hyperbolic 3-manifolds, namely those which are obtained as n-fold branched covers of hyperbolic knots in S3. Let S(K,α) denote the hyperbolic cone manifold whose underlying topological space is S3 and whose singular locus is a geodesic isotopic to K with cone angle α. Such a manifold is obtained, for instance, by (generalized) hyperbolic Dehn surgery of type (rp,rq) on K with p,q relatively prime integers, and r real. If such a geometric object exists, it has cone angle 2rπ. Thurston's "orbifold theorem'' implies that if r≤1/2 and K satisfies certain topological conditions, then the cone manifold in question exists. Then the n-fold orbifold cover of S(K,2π/n) is the hyperbolic manifold Mn(K) obtained by n-fold cyclic covering of S3 branched over the knot K. The well-known "Schläfli formula'' says that the derivative of the volume of S(K,α) as a function of α is proportional to the length of the singular curve. The authors establish that the derivative of a generalization of the Chern-Simons invariant of S(K,α) as a function of α is proportional to the "jump'' of the singular curve—a generalization of the notion of torsion for a geodesic in a nonsingular hyperbolic manifold. This is proved by using the analyticity of the relevant quantities as functions on the PSL(2,C) representation variety of π1(S3−K), a result due to T. Yoshida [Invent. Math. 81 (1985), no. 3, 473–514;]. This together with the multiplicative nature of the Chern-Simons invariant for orbifolds allows one to calculate the Chern-Simons invariant of Mn(K).