On the character variety of tunnel number 1 knots
| dc.contributor.author | Hilden, Hugh Michael | |
| dc.contributor.author | Lozano Imízcoz, María Teresa | |
| dc.contributor.author | Montesinos Amilibia, José María | |
| dc.date.accessioned | 2023-06-20T18:47:53Z | |
| dc.date.available | 2023-06-20T18:47:53Z | |
| dc.date.issued | 2000 | |
| dc.description.abstract | Given a hyperbolic knot K in S3, the SL2(C) characters ofπ1(S3−K) form an algebraic variety Cˆ(K). The algebraic component containing the character of the complete hyperbolic structure of S3−K is an algebraic curve CˆE(K). The desingularization of the projective curve corresponding to CˆE(K) is a Riemann surface Σ(K), and the trace function corresponding to the meridian of K induces a map p:Σ(K)→C. The pair (Σ(K),p) contains a great deal of information about the knot K and its hyperbolic structure. It can be described by a polynomial rE[K](y,z). There is an algebraic number yh which is a particular critical point of p in the interval (−2,2). It defines an angle 0<αh<2π with yh=2cos(αh/2), called the limit of hyperbolicity. The minimal polynomial hK(y) of yh is called the h-polynomial of K. The calculation of these invariants is in general quite complicated. In this paper the authors develop a method to calculate rE[K](y,z) and hK(y) for any tunnel number one knot, and they apply the method to the knots 10139 and 10161. | |
| dc.description.department | Depto. de Álgebra, Geometría y Topología | |
| dc.description.faculty | Fac. de Ciencias Matemáticas | |
| dc.description.refereed | TRUE | |
| dc.description.status | pub | |
| dc.eprint.id | https://eprints.ucm.es/id/eprint/22237 | |
| dc.identifier.doi | 10.1112/S0024610700001605 | |
| dc.identifier.issn | 0024-6107 | |
| dc.identifier.officialurl | http://jlms.oxfordjournals.org/content/62/3/938 | |
| dc.identifier.relatedurl | http://www.cambridge.org/ | |
| dc.identifier.uri | https://hdl.handle.net/20.500.14352/58646 | |
| dc.issue.number | 3 | |
| dc.journal.title | Journal of the London Mathematical Society. Second Series | |
| dc.page.final | 950 | |
| dc.page.initial | 938 | |
| dc.publisher | Oxford University Press | |
| dc.relation.projectID | PB95-0413 | |
| dc.relation.projectID | PB98-0826. | |
| dc.rights.accessRights | metadata only access | |
| dc.subject.cdu | 515.162.8 | |
| dc.subject.keyword | periodic links | |
| dc.subject.ucm | Topología | |
| dc.subject.unesco | 1210 Topología | |
| dc.title | On the character variety of tunnel number 1 knots | |
| dc.type | journal article | |
| dc.volume.number | 62 | |
| dcterms.references | M. BOILEAU and J. PORTI, `Geometrization of 3-orbifolds of cyclic type', AsteT risque (Socie! te! Mathe! matique de France, Paris, to appear). cf. G. BRUMFIEL and H. HILDEN, `SL(2)-representations of finitely presented groups', Contemp. Math. 187 (1995). G. BURDE, `SU(2)-representation spaces for two-bridge knot groups', Math. Ann. 173 (1990) 103–119. A. CASSA, Teoria elementare delle cur e algebraiche piane e delle superfici di Riemann compatte (Pitagora Editrice, Bolonia, 1983). D. COOPER, M. CULLER, H. GILLET, D. D. LONG and P. B. SHALEN, `Plane curves associated to character varieties of 3-manifolds', In ent. Math. 118 (1994) 47–84. M. CULLER, C. M. GORDON, J. LUECKE and P. B. SHALEN, `Dehn surgery on knots', Ann. of Math. 125 (1987) 237–300. M. CULLER and P. B. SHALEN, `Varieties of group representations and splitting of 3-manifolds', Ann. of Math. 117 (1983) 109–146. F. GONZA! LEZ-ACUN4 A and J. M. MONTESINOS, `On the character variety of group representations in SL(2, ) and PSL(2, )', Math. Z. 214 (1993) 627–652. M. HEUSENER, `SO(R)-representation curves for two-bridge knot groups', Math. Ann. 298 (1994) 327–348. H. M. HILDEN, M. T. LOZANO and J. M. MONTESINOS-AMILIBIA, `On the character variety of group representations of a 2-bridge link p∖3 into PSL(2, )', Bull. Soc. Mat. Mexicana 37 (1992) 241–262. H. M. HILDEN, M. T. LOZANO and J. M. MONTESINOS-AMILIBIA, `On the arithmetic 2-bridge knots and link orbifolds and a new knot invariant', J. Knot Theory Ramifications 4 (1995) 81–114. H. M. HILDEN, M. T. LOZANO and J. M. MONTESINOS-AMILIBIA, `On volumes and Chern–Simons invariants of geometric 3-manifolds', J. Math. Sci. Uni. Tokyo 3 (1996) 723–744. H. M. HILDEN, M. T. LOZANO and J. M. MONTESINOS-AMILIBIA, `Volumes and Chern–Simons invariants of cyclic coverings over rational knots', Topology and TeichmuW ller spaces (ed. S. Kojima et al., World Scientific, 1996) 31–55. H. M. HILDEN, M. T. LOZANO and J. M. MONTESINOS-AMILIBIA, `The Chern–Simons invariants of hyperbolic manifolds via covering spaces', Bull. London Math. Soc. 31 (1999) 354–366. H. M. HILDEN, M. T. LOZANO and J. M. MONTESINOS-AMILIBIA, `On the character variety of periodic knots and links', Math. Proc. Cambridge Philos. Soc. (to appear). cf. H. M. HILDEN, M. T. LOZANO and J. M. MONTESINOS-AMILIBIA, `Character varieties of (2, n)-mark knots', preprint, 1999. cf. E. KLASSEN, `Representations of knot groups in SU(2)', Trans. Amer. Math. Soc. 326 (1991) 795–828. S. KOJIMA, `Nonsingular parts of hyperbolic 3-cone-manifolds', Topology and TeichmuW ller spaces (ed. S. Kojima et al., World Scientific, 1996) 115–122. M. T. LOZANO and J. M. MONTESINOS-AMILIBIA, `Geodesic flows on hyperbolic orbifolds, and universal orbifolds', Pacific J. Math. 177 (1997) 109–147. J. M. MONTESINOS, `Surgery on links and double branched covers of S', Knots, groups and 3- manifolds, Annals of Mathematics Studies 84 (ed. L. P. Neuwirth, Princeton University Press, 1975) 227–259. J. M. MONTESINOS and W. WHITTEN, `Constructions of two-fold branched covering spaces', Pacific J. Math. 125 (1986) 415–446. K. MORIMOTO, M. SAKUMA and Y. YOKOTA, `Identifying tunnel number one knots', J. Math. Soc. Japan 48 (1996) 667–688. T. OHTSUKI, `Ideal points and incompressible surfaces in two-bridge knot complements', J. Math. Soc. Japan 46 (1994) 51–87. R. RILEY, `Seven excellent knots', Low dimensional topology, London Mathematical Society Lecture Note Series 48 (ed. R. Brown and T. L. Thickstun, Cambridge University Press, 1982) 81–151. R. RILEY, `Non-abelian representations of 2-bridge knot groups', Quart. J. Math. Oxford Ser. (2) 35 (1984) 191–208. R. RILEY, `Algebra for Heckoid groups', Trans. Amer. Math. Soc. 334 (1992) 389–409. D. ROLFSEN, Knots and links (Publish or Perish, 1976). P. SCOTT, `The geometries of 3-manifolds', Bull. London Math. Soc. 15 (1983) 401–487. W. THURSTON, `The geometry and topology of 3-manifolds' (Princeton University Press, Princeton, to appear) (notes 1976–78). A. WHITEMORE, `On representations of the group of Listing's knot by subgroups of PSL(2, )', Proc. Amer. Math. Soc. 40 (1973) 378–382. | |
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