Hyperbolic covering knots

Silver, Daniel S¹ ; Whitten, Wilbur²

¹Department of Mathematics, University of South Alabama, Mobile AL 36688, USA
²1620 Cottontown Road, Forest VA 24551, USA

Algebraic and Geometric Topology, Tome 5 (2005) no. 4, pp. 1451-1469

Résumé

Given any knot k, there exists a hyperbolic knot k̃ with arbitrarily large volume such that the knot group πk is a quotient of πk̃ by a map that sends meridian to meridian and longitude to longitude. The knot k̃ can be chosen to be ribbon concordant to k and also to have the same Alexander invariant as k.

Détail

DOI : 10.2140/agt.2005.5.1451

Keywords: Alexander module, hyperbolic knot, ribbon concordance, tangle

Bibliographie

[1] S A Bleiler, Knots prime on many strings, Trans. Amer. Math. Soc. 282 (1984) 385

[2] G Burde, K Murasugi, Links and Seifert fiber spaces, Duke Math. J. 37 (1970) 89

[3] M Culler, C M Gordon, J Luecke, P B Shalen, Dehn surgery on knots, Ann. of Math. $(2)$ 125 (1987) 237

[4] S Eliahou, L H Kauffman, M B Thistlethwaite, Infinite families of links with trivial Jones polynomial, Topology 42 (2003) 155

[5] C M Gordon, Ribbon concordance of knots in the 3–sphere, Math. Ann. 257 (1981) 157

[6] C M Gordon, R A Litherland, Incompressible surfaces in branched coverings, from: "The Smith conjecture (New York, 1979)", Pure Appl. Math. 112, Academic Press (1984) 139

[7] M Gromov, Volume and bounded cohomology, Inst. Hautes Études Sci. Publ. Math. (1982)

[8] D Johnson, C Livingston, Peripherally specified homomorphs of knot groups, Trans. Amer. Math. Soc. 311 (1989) 135

[9] E Kalfagianni, Alexander polynomial, finite type invariants and volume of hyperbolic knots, Algebr. Geom. Topol. 4 (2004) 1111

[10] A Kawauchi, Almost identical link imitations and the skein polynomial, from: "Knots 90 (Osaka, 1990)", de Gruyter (1992) 465

[11] A Kawauchi, Almost identical imitations of $(3,1)$–dimensional manifold pairs, Osaka J. Math. 26 (1989) 743

[12] P K Kim, J L Tollefson, Splitting the PL involutions of nonprime 3–manifolds, Michigan Math. J. 27 (1980) 259

[13] P B Kronheimer, T S Mrowka, Witten's conjecture and property P, Geom. Topol. 8 (2004) 295

[14] W B R Lickorish, Prime knots and tangles, Trans. Amer. Math. Soc. 267 (1981) 321

[15] C Livingston, Computations of the Ozsváth–Szabó knot concordance invariant, Geom. Topol. 8 (2004) 735

[16] K Miyazaki, Ribbon concordance does not imply a degree one map, Proc. Amer. Math. Soc. 108 (1990) 1055

[17] R Myers, Simple knots in compact, orientable 3–manifolds, Trans. Amer. Math. Soc. 273 (1982) 75

[18] Y Nakanishi, Prime links, concordance and Alexander invariants, Math. Sem. Notes Kobe Univ. 8 (1980) 561

[19] Y Nakanishi, Primeness of links, Math. Sem. Notes Kobe Univ. 9 (1981) 415

[20] Y Nakanishi, Prime and simple links, Math. Sem. Notes Kobe Univ. 11 (1983) 249

[21] J Rasmussen, Khovanov homology and the slice genus

[22] L Rudolph, Quasipositivity as an obstruction to sliceness, Bull. Amer. Math. Soc. $($N.S.$)$ 29 (1993) 51

[23] D S Silver, W Whitten, Knot group epimorphisms, J. Knot Theory Ramifications 15 (2006) 153

[24] A Stoimenow, On polynomials and surfaces of variously positive links, J. Eur. Math. Soc. $($JEMS$)$ 7 (2005) 477

[25] W P Thurston, Three-dimensional geometry and topology Vol. 1, Princeton Mathematical Series 35, Princeton University Press (1997)

[26] W P Thurston, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. $($N.S.$)$ 6 (1982) 357

[27] S Wang, Non-zero degree maps between 3–manifolds, from: "Proceedings of the International Congress of Mathematicians, Vol II (Beijing, 2002)", Higher Ed. Press (2002) 457