logo AFST
Knot complements, hidden symmetries and reflection orbifolds
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 24 (2015) no. 5, pp. 1179-1201.

Dans cet article, nous étudions la conjecture de Neumann et Reid selon laquelle les seuls nœuds hyperboliques dans la sphère S 3 admettant des symétries cachées sont le nœud figure-huit et les deux nœuds dodécaédriques. Les nœuds dont les compléments revêtent des orbifold de réflexions hyperboliques admettent des symétries cachées et nous vérifions la conjecture de Neumann et Reid pour ces nœuds lorsque l’orbifold de réflexions est petit. Nous montrons aussi qu’un orbifold de réflexions revêtu par le complément d’un nœud “AP” est nécessairement petit. Ainsi, lorsque K est un nœud “AP”, le complément de K revêt un orbifold de réflexions si et seulement si K est le nœud figure-huit ou l’un des deux nœuds dodécaédriques.

In this article we examine the conjecture of Neumann and Reid that the only hyperbolic knots in the 3-sphere which admit hidden symmetries are the figure-eight knot and the two dodecahedral knots. Knots whose complements cover hyperbolic reflection orbifolds admit hidden symmetries, and we verify the Neumann-Reid conjecture for knots which cover small hyperbolic reflection orbifolds. We also show that a reflection orbifold covered by the complement of an AP knot is necessarily small. Thus when K is an AP knot, the complement of K covers a reflection orbifold exactly when K is either the figure-eight knot or one of the dodecahedral knots.

Publié le :
DOI : https://doi.org/10.5802/afst.1480
@article{AFST_2015_6_24_5_1179_0,
     author = {Michel Boileau and Steven Boyer and Radu Cebanu and Genevieve S. Walsh},
     title = {Knot complements, hidden symmetries and reflection orbifolds},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {1179--1201},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 24},
     number = {5},
     year = {2015},
     doi = {10.5802/afst.1480},
     language = {en},
     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1480/}
}
TY  - JOUR
AU  - Michel Boileau
AU  - Steven Boyer
AU  - Radu Cebanu
AU  - Genevieve S. Walsh
TI  - Knot complements, hidden symmetries and reflection orbifolds
JO  - Annales de la Faculté des sciences de Toulouse : Mathématiques
PY  - 2015
DA  - 2015///
SP  - 1179
EP  - 1201
VL  - Ser. 6, 24
IS  - 5
PB  - Université Paul Sabatier, Toulouse
UR  - https://afst.centre-mersenne.org/articles/10.5802/afst.1480/
UR  - https://doi.org/10.5802/afst.1480
DO  - 10.5802/afst.1480
LA  - en
ID  - AFST_2015_6_24_5_1179_0
ER  - 
%0 Journal Article
%A Michel Boileau
%A Steven Boyer
%A Radu Cebanu
%A Genevieve S. Walsh
%T Knot complements, hidden symmetries and reflection orbifolds
%J Annales de la Faculté des sciences de Toulouse : Mathématiques
%D 2015
%P 1179-1201
%V Ser. 6, 24
%N 5
%I Université Paul Sabatier, Toulouse
%U https://doi.org/10.5802/afst.1480
%R 10.5802/afst.1480
%G en
%F AFST_2015_6_24_5_1179_0
Michel Boileau; Steven Boyer; Radu Cebanu; Genevieve S. Walsh. Knot complements, hidden symmetries and reflection orbifolds. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 24 (2015) no. 5, pp. 1179-1201. doi : 10.5802/afst.1480. https://afst.centre-mersenne.org/articles/10.5802/afst.1480/

[1] Adams (C.).— Toroidally alternating knots and links, Topology 33, p. 353-369 (1994). | MR 1273788 | Zbl 0839.57004

[2] Aitchison (I. R.) and Rubinstein (J. H.).— Combinatorial cubings, cusps, and the dodecahedral knots, in Topology ’90, Ohio State Univ. Math. Res. Inst. Publ. 1, p. 17-26, de Gruyter (1992). | MR 1184399 | Zbl 0773.57010

[3] Aitchison (I. R.) and Rubinstein (J. H.).— Geodesic Surfaces in Knot Complements, Exp. Math. 6, p. 137-150 (1997). | EuDML 226361 | MR 1474574 | Zbl 0891.57017

[4] Banks (J.).— The complement of a dodecahedral knot contains an essential closed surface, , preprint 2012.

[5] Boileau (M.), Boyer (S.), Cebanu (R.), and Walsh (G. S.).— Knot commensurability and the Berge conjecture, Geom. & Top. 16, p. 625-664 (2012). | MR 2928979 | Zbl 1258.57001

[6] Boileau (M.), Maillot (S.) and Porti (J.).— Three-dimensional orbifolds and their geometric structures, Panoramas et Synthèses, 15, Société Mathématique de France, Paris (2003). | MR 2060653 | Zbl 1058.57009

[7] Burton (B.), Coward (A.), and Tillmann (S.).— Computing closed essential surfaces in knot complements, SoCG ’13: Proceedings of the Twenty-Ninth Annual Symposium on Computational Geometry, ACM, p. 405-414 (2013). | MR 3208239 | Zbl 1305.68213

[8] Dolgachev (I.).— Reflection groups in algebraic geometry, Bull. Amer. Math. Soc. 45, p. 1-60 (2008). | Zbl 1278.14001

[9] González-Acuña (F.) and Whitten (W. C.).— Imbeddings of three-manifold groups, Mem. Amer. Math. Soc. 474 (1992). | MR 1117167 | Zbl 0756.57002

[10] Goodman (O.), Heard (D.), and Hodgson (C.).— Commensurators of cusped hyperbolic manifolds, Experiment. Math. 17, p. 283-306 (2008). | MR 2455701

[11] Hoffman (N.).— Small knot complements, exceptional surgeries, and hidden symmetries, Alg. & Geom. Top. 14, p. 3227-3258 (2014).

[12] Hoffman (N.).— On knot complements that decompose into regular ideal dodecahedra, to appear in Geom. Dedicata. | Zbl 1305.57037

[13] Jaco (Wm.) and Shalen (P. B.).— Seifert fibered spaces in 3-manifolds, Mem. Amer. Math. Soc. 21, no. 220 (1979). | Zbl 0415.57005

[14] Maclachlan (C.) and Reid (A. W.).— The Arithmetic of Hyperbolic 3-manifolds, Graduate Texts in Mathematics 219, Springer-Verlag (2003). | MR 1937957 | Zbl 1025.57001

[15] Margulis (G.).— Discrete Subgroups of Semi-simple Lie Groups, Ergeb. der Math. 17 Springer-Verlag (1989). | Zbl 0732.22008

[16] Menasco (Wm.).— Closed incompressible surfaces in alternating knot and link complements, Top. 23, p. 7-44 (1984). | Zbl 0525.57003

[17] Neumann (W. D.) and Reid (A. W.).— Arithmetic of hyperbolic manifolds, in Topology ’90, Ohio State Univ. Math. Res. Inst. Publ. 1, p. 273-310, de Gruyter (1992). | MR 1184416 | Zbl 0747.00024

[18] Neumann (W. D.) and Reid (A. W.).— Notes on Adams’ small volume orbifolds, in Topology ’90, Ohio State Univ. Math. Res. Inst. Publ. 1, p. 273-310, de Gruyter (1992). | MR 1184416 | Zbl 0747.00024

[19] Oertel (U.).— Closed incompressible surfaces in complements of star links, Pac. J. Math, 111, p. 209-230 (1984). | MR 732067 | Zbl 0549.57004

[20] Reid (A. W.).— Arithmeticity of knot complements, J. London Math. Soc. 43, p. 171-184 (1991). | MR 1099096 | Zbl 0847.57013

[21] Reid (A. W.) and Walsh (G. S.).— Commensurability classes of two-bridge knot complements, Alg. & Geom. Top. 8, p. 1031-1057 (2008). | MR 2443107 | Zbl 1154.57001

[22] Schwartz (R; E.).— The quasi-isometry classification of rank one lattices, Publ. I.H.E.S. 82, p. 133-168 (1995). | Numdam | MR 1383215 | Zbl 0852.22010

[23] Thurston (Wm.).— The Geometry and Topology of 3-manifolds, Princeton University lecture notes, 1980. Electronic version 1.1: http://www.msri.org/publications/books/gt3m/.

[24] Walsh (G. S.).— Orbifolds and commensurability, In “Interactions Between Hyperbolic Geometry, Quantum Topology and Number Theory" Contemporary Mathematics 541, p. 221-231 (2011). | MR 2796635 | Zbl 1231.57017

[25] Wielenberg (N.).— The structure of certain subgroups of the Picard group, Math. Proc. Cam. Phil. Soc. 84, p. 427-436 (1978). | MR 503003 | Zbl 0399.57005

Cité par Sources :