Cet article est consacré à un début d’énumération des compléments d’entrelacs dans provenant des groupes de congruence principaux. Nous utilisons des méthodes théoriques ainsi que des calculs avec MAGMA.
In this paper we study principal congruence link complements in . It is known that there are only finitely many such link complements, and we make a start on enumerating them using a combination of theoretical methods and computer calculations with MAGMA.
@article{AFST_2014_6_23_5_1063_0, author = {Mark D. Baker and Alan W. Reid}, title = {Principal congruence link complements}, journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques}, pages = {1063--1092}, publisher = {Universit\'e Paul Sabatier, Institut de Math\'ematiques}, address = {Toulouse}, volume = {Ser. 6, 23}, number = {5}, year = {2014}, doi = {10.5802/afst.1436}, mrnumber = {3294602}, language = {en}, url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1436/} }
TY - JOUR AU - Mark D. Baker AU - Alan W. Reid TI - Principal congruence link complements JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2014 SP - 1063 EP - 1092 VL - 23 IS - 5 PB - Université Paul Sabatier, Institut de Mathématiques PP - Toulouse UR - https://afst.centre-mersenne.org/articles/10.5802/afst.1436/ DO - 10.5802/afst.1436 LA - en ID - AFST_2014_6_23_5_1063_0 ER -
%0 Journal Article %A Mark D. Baker %A Alan W. Reid %T Principal congruence link complements %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2014 %P 1063-1092 %V 23 %N 5 %I Université Paul Sabatier, Institut de Mathématiques %C Toulouse %U https://afst.centre-mersenne.org/articles/10.5802/afst.1436/ %R 10.5802/afst.1436 %G en %F AFST_2014_6_23_5_1063_0
Mark D. Baker; Alan W. Reid. Principal congruence link complements. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Numéro Spécial : Aux croisements de la géométrie hyperbolique et de l’arithmétique, Tome 23 (2014) no. 5, pp. 1063-1092. doi : 10.5802/afst.1436. https://afst.centre-mersenne.org/articles/10.5802/afst.1436/
[1] Adams (C. C.) and Reid (A. W.).— Systoles of hyperbolic 3-manifolds, Math. Proc. Camb. Phil. Soc. 128, p. 103-110 (2000). | MR | Zbl
[2] Agol (I.).— Bounds on exceptional Dehn filling, Geometry and Topology 4, p. 431-449 (2000). | MR | Zbl
[3] Baker (M. D.).— Link complements and quadratic imaginary number fields, Ph.D Thesis M.I.T. (1981).
[4] Baker (M. D.).— Link complements and the homology of arithmetic subgroups of , I.H.E.S. preprint (1982).
[5] Baker (M. D.).— Link complements and the Bianchi modular groups, Trans. A. M. S. 353, p. 3229-3246 (2001). | MR | Zbl
[6] Baker (M. D.).— Link complements and integer rings of class number greater than one, TOPOLOGY ’90 p. 55-59, Ohio State Univ. Math. Res. Inst. Publ., 1, de Gruyter (1992). | MR | Zbl
[7] Baker (M. D.) and Reid (A. W.).— Arithmetic knots in closed 3-manifolds, in Proceedings of Knots 2000, J. Knot Theory and its Ramifications 11, p. 903-920 (2002). | MR | Zbl
[8] Bosma (W.), Cannon (J.), and Playoust (C.).— The Magma algebra system. I. The user language, J. Symbolic Comput. 24, p. 235-265 (1997). | MR | Zbl
[9] Coulsen (D.), Goodman (O. A.), Hodgson (C. D.) and Neumann (W. D.).— Computing arithmetic invariants of 3-manifolds, Experimental J. Math. 9, p. 127-152 (2000). | MR | Zbl
[10] Culler (M.), Dunfield (N.) and Weeks (J.).— SnapPy, a computer program for studying the geometry and topology of 3-manifolds, http://snappy.computop.org.
[11] Dennin (J. B.).— Fields of modular functions of genus , Illinois J. Math. 15, p. 442-455 (1971). | MR | Zbl
[12] Dennin (J. B.).— Subfields of of genus , Illinois J. Math. 16, p. 502-518 (1972). | MR | Zbl
[13] Dennin (J. B.).— The genus of subfields of , Illinois J. Math. 18, p. 246-264 (1974). | MR | Zbl
[14] Dixon (L. E.).— Linear groups, with an exposition of the Galois field theory, Dover Publications, (1958). | MR
[15] Futer (D.), Kalfagianni (E.) and Purcell (J.).— On diagrammatic bounds of knot volumes and spectral invariants, Geom. Dedicata 147, p. 115-130 (2010). | MR | Zbl
[16] Goerner (M.).— Visualizing regular tessellations: principal congruence links and and equivariant morphisms from surfaces to 3-manifold, Ph.D Thesis, U. C. Berkeley (2011). | MR
[17] Goerner (M.).— Regular tessellation links, ArXiv:1406.2827.
[18] Gordon (C. McA.).— Links and their complements, in Topology and Geometry: commemorating SISTAG, p. 71-82, Contemp. Math., 314, Amer. Math. Soc. (2002). | MR | Zbl
[19] Grunewald (F.) and Schwermer (J.).— Arithmetic quatients of hyperbolic 3-space, cusp forms and link complements, Duke Math. J. 48, p. 351-358 (1981). | MR | Zbl
[20] Grunewald (F.) and Schwermer (J.).— A non-vanishing theorem for the cuspidal cohomology of over imaginary quadratic integers, Math. Annalen 258, p. 183-200 (1981). | MR | Zbl
[21] Grunewald (F.) and Schwermer (J.).— Subgroups of Bianchi groups and arithmetic quotients of hyperbolic -space, Trans A. M. S. 335, p. 47-78 (1993). | MR | Zbl
[22] Hatcher (A.).— Hyperbolic structures of arithmetic type on some link complements, J. London Math. Soc 27, p. 345-355 (1983). | MR | Zbl
[23] Lackenby (M.).— Word hyperbolic Dehn surgery, Invent. Math. 140, p. 243-282 (2000). | MR | Zbl
[24] Lackenby (M.).— Spectral geometry, link complements and surgery diagrams, Geom. Dedicata 147, p. 191-206 (2010). | MR | Zbl
[25] Lakeland (G.) and Leininger (C.).— Systoles and Dehn surgery for hyperbolic 3-manifolds , to appear Algebraic and Geometric Topology. | MR | Zbl
[26] Maclachlan (C.) and Reid (A. W.).— The Arithmetic of Hyperbolic 3-Manifolds, Graduate Texts in Mathematics, 219, Springer-Verlag (2003). | MR | Zbl
[27] Reid (A. W.).— The arithmeticity of knot complements, J. London Math. Soc. 43, p. 171-184 (1991). | MR | Zbl
[28] Sebbar (A.).— Classification of torsion-free genus zero congruence subgroups, Proc. A. M. S. 129, p. 2517-2527 (2001). | MR | Zbl
[29] Shimura (G.).— Introduction to the Arithmetic Theory of Automorphic Functions, Publications of the Math. Society of Japan 11 (1971). | MR | Zbl
[30] Swan (R. G.).— Generators and relations for certain special linear groups, Advances in Math. 6, p. 1-77 (1971). | MR | Zbl
[31] Thompson (J. G.).— A finiteness theorem for subgroups of , in Proc. Symp. Pure Math. 37, 533-555, A.M.S. Publications (1980). | MR | Zbl
[32] Thurston (W. P.).— The Geometry and Topology of 3-Manifolds, Princeton University mimeographed notes, (1979).
[33] Thurston (W. P.).— Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. A. M. S. 6, p. 357-381 (1982). | MR | Zbl
[34] Vogtmann (K.).— Rational homology of Bianchi groups, Math. Ann. 272, p. 399-419 (1985). | MR | Zbl
[35] Zograf (P.).— A spectral proof of Rademacher’s conjecture for congruence subgroups of the modular group, J. Reine Angew. Math. 414, p. 113-116 (1991). | MR | Zbl
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