Peripheral polynomials of hyperbolic knots

Aviso: antes de depositar, asegúrate que el documento no esta ya en el repositorio. Para ello busca por el título: escribe solo la primera parte y excluye los dos puntos, guiones, etc. y toda la frase posterior. Para depositar documentos, por favor, inicia sesión e identifícate con tu cuenta de correo institucional de la UCM con el botón MI CUENTA UCM. No emplees la opción AUTENTICACIÓN CON CONTRASEÑA

Publication:
Peripheral polynomials of hyperbolic knots

Files

montesinos69.pdf (222.59 KB)

Official URL

http://www.sciencedirect.com/science/article/pii/S0166864104003840

Publication Date

2005

Authors

Hilden, Hugh Michael

Lozano Imízcoz, María Teresa

Montesinos Amilibia, José María

Publisher

Elsevier Science

Citations

Exportar

Abstract

If K is a hyperbolic knot in S3, an algebraic component of its character variety containing one holonomy of the complete hyperbolic structure of finite volume of S3∖K is an algebraic curve K. The traces of the peripheral elements of K define polynomial functions in K, which are related in pairs by polynomials (peripheral polynomials). These are determined by just two adjacent peripheral polynomials. The curves defined by the peripheral polynomials are all birationally equivalent to K, with only one possible exception. The canonical peripheral polynomial relating the trace of the meridian with the trace of the canonical longitude of K is a factor of the A-polynomial.

UCM subjects

Geometria algebraica , Topología

Unesco subjects

1201.01 Geometría Algebraica , 1210 Topología

Citation

G. Burde, SU (2)-representation spaces for two-bridge knot groups, Math. Ann. 173 (1990) 103–119. A. Cassa, Teoria Elementare delle Curve Algebriche Piane e delle Superfici de Riemann Compatte, Quaderni dell'Unione Matematica Italiana, vol. 25, Pitagora Editrice, Bologna, 1983. D. Cox, J. Little, D. O'Shea, Ideals, Varieties, and Algorithms, Springer, Berlin, 1992. MR1189133 (93j:13031) M. Culler, Lifting representations to covering groups, Adv. Math. 59 (1986) 64–70. D. Cooper, D.D. Long, Remarks on the A-polynomial of a knot, J. Knot Theory Ramifications 5 (1996) 609–628. M. Culler, P.B. Shalen, Varieties of group representations and splitting of 3-manifolds, Ann. of Math. 117 (1983) 109–146. M. Culler, P.B. Shalen, Bounded, separating surfaces in knot manifolds, Invent. Math. 75 (1984) 537–545. N.M. Dunfield, Cyclic surgery, degree of maps of character curves, and volume rigidity for hyperbolic manifolds, Invent. Math. 136 (1999) 623–657. F. González-Acu~na, J.M. Montesinos, On the character variety of group representations in SL(2,C) and PSL(2,C), Math. Z. 214 (1993) 627–652. A. Hatcher, W. Thurston, Incompressible surfaces in 2-bridge knot complements, Invent. Math. 79 (1985) 225–246. M. Heusener, A polynomial invariant via representation spaces, J. Knots Theory Ramifications 3 (1994) 11–23. H.M. Hilden, M.T. Lozano, J.M. Montesinos-Amilibia, The arithmeticity of the figure eight knot orbifolds, in: B. Apanasov, W. Neumann, A. Reid, L. Siebebenmann (Eds.), Topology '90, in: Ohio State Univ. Math. Res. Inst. Pub., vol. 1, de Gruyter, Berlin, 1992, pp. 169–183. H.M. Hilden, M.T. Lozano, J.M. Montesinos-Amilibia, On the character variety of group representations of a 2-bridge link p/3 into PSL(2,C), Bol. Soc. Mat. Mexicana 37 (1992) 241–262. H.M. Hilden, M.T. Lozano, J.M. Montesinos-Amilibia, On a remarkable polyhedron geometrizing the figure-eight knot cone manifolds, J. Math. Sci. Uni. Tokyo 2 (1995) 501–561. H.M. Hilden, M.T. Lozano, J.M. Montesinos-Amilibia, On the arithmetic 2-bridge knots and link orbifolds and a new knot invariant, J. Knots Theory Ramifications 4 (1995) 81–114. H.M. Hilden, M.T. Lozano, J.M. Montesinos-Amilibia, On volumes and Chern-Simons invariants of geometric 3-manifolds, J. Math. Sci. Univ. Tokyo 3 (1996) 723–744. H.M. Hilden, M.T. Lozano, J.M. Montesinos-Amilibia, Volumes and Chern-Simons invariants of cyclic coverings over rational knots, in: S. Kojima, et al. (Eds.), Topology and Teichmüller Spaces, World Scientific, Singapore, 1996, pp. 31–55. H.M. Hilden, M.T. Lozano, J.M. Montesinos-Amilibia, The arithmeticity of certain torus bundle cone 3-manifolds and hyperbolic surface bundle 3-manifolds and an enhanced arithmeticity test, in: S. Suzuki (Ed.), Proc. of Knots '96, World Scientific, Singapore, 1997, pp. 73–80. H.M. Hilden, M.T. Lozano, 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, J.M. Montesinos-Amilibia, On the character variety of periodic knots and links, Math. Proc. Cambridge Philos. Soc. 129 (2000) 477–490. H.M. Hilden, M.T. Lozano, J.M. Montesinos-Amilibia, Character varieties of tunnel number one knots, J. London Math. Soc. (2) 62 (2000) 938–950. H.M. Hilden, M.T. Lozano, J.M. Montesinos-Amilibia, Character varieties and perhipheral polynomials of a class of knots, J. Knots Theory Ramifications 12 (8) (2003) 1093–1130. C.D. Hodgson, Degeneration and regeneration of geometric structures on three-manifolds, Ph.D. Thesis, Princeton University, 1986. R. Riley, Algebra for Heckoid groups, Trans. Amer. Math. Soc. 334 (1992) 389–409. W. Thurston, The geometry and topology of 3-manifolds, Notes 1976–1978, http://www.msri.org/publications/books/gt3m/. H. Vogt, Sur les invariants fondamentaux des equations differentielles lineaires de second ordre, Ann. Sci. Ecole Norm. Sup. (3) 6 (Suppl.) (1889) 3–72. A. Whittemore, On representations of the group of Listing's knot by subgroups of SL(2,C), Proc. Amer. Math. Soc. 40 (1973) 378–382. S. Wolfram, Mathematica

URI

https://hdl.handle.net/20.500.14352/50759

Collections