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.
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.
@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, Serie 6, Volume 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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