No Access

Bridge spectra of cables of 2-bridge knots

Okinawa Institute of Science and Technology Graduate University, 1919-1 Tancha, Onna-son, Kunigami-gun, Okinawa, Japan 904-0495, Japan

E-mail Address: nick@owad.org

Search for more papers by this author

https://doi.org/10.1142/S0218216518500128Cited by:0 (Source: Crossref)

Abstract

We compute the bridge spectra of cables of 2-bridge knots. We also give some results about bridge spectra and distance of Montesinos knots.

Keywords:

AMSC: 57M25, 57M27

References

1. D. Bachman and S. Schleimer, Distance and bridge position, Pacific J. Math. 219(2) (2005) 221–235. Crossref, Web of Science, Google Scholar
2. M. Boileau and H. Zieschang, Nombre de ponts et générateurs méridiens des entrelacs de Montesinos, Comment. Math. Helv. 60 (1985) 270–279. Crossref, Web of Science, Google Scholar
3. S. Bowman, S. Taylor and A. Zupan, Bridge spectra of twisted torus knots, Int. Math. Res. Not. 16 (2015) 7336–7356. Crossref, Web of Science, Google Scholar
4. H. Doll, A generalized bridge number for links in 3-manifolds, Math. Ann. 294(4) (1992) 701–717. Crossref, Web of Science, Google Scholar
5. C. McA. Gordon and R. A. Litherland, Incompressible planar surfaces in 3-manifolds, Topology Appl. 18(2–3) (1984) 121–144. Crossref, Google Scholar
6. A. Hatcher, Notes on basic 3-manifold topology, book in progress (2007). Google Scholar
7. A. Hatcher and W. Thurston, Incompressible surfaces in 2-bridge knot complements, Invent. Math. 79 (1985) 225–246. Crossref, Web of Science, Google Scholar
8. C. Hayashi and K. Shimokawa, Thin position of a pair (3-manifold, 1-submanifold), Pacific J. Math. 197 (2001) 301–324. Crossref, Web of Science, Google Scholar
9. M. Hirasawa and K. Murasugi, Genera and fibredness of Montesinos knots, Pacific J. Math. 225(1) (2006) 53–83. Crossref, Web of Science, Google Scholar
10. L. H. Kauffman and S. Lambropoulou, On the classification of rational tangles, Adv. Appl. Math. 33(2) (2004) 199–237. Crossref, Web of Science, Google Scholar
11. M. Lustig and Y. Moriah, Generalized Montesinos knots and $N$ -torsion, Math. Ann. 295 (1993) 167–189. Crossref, Web of Science, Google Scholar
12. J. M. Montesinos, Variedades de Seifert que son recubridadores ciclicols ramificados de dos hojas, Bol. Soc. Mat. Mexicana 18 (1973) 1–32. Google Scholar
13. Y. Moriah, A note on satellites and tunnel number, Kobe J. Math. 8 (1991) 73–79. Web of Science, Google Scholar
14. K. Morimoto, On composite types of tunnel number two knots, J. Knot Theory Ramifications 24(2) (2015) 1550013. Link, Web of Science, Google Scholar
15. K. Morimoto and M. Sakuma, On unknotting tunnels for knots, Math. Ann. 289 (1991) 143–167. Crossref, Web of Science, Google Scholar
16. K. Morimoto, M. Sakuma and Y. Yokota, Identifying tunnel number one knots, J. Math. Soc. Japan 48(4) (1996) 667–688. Crossref, Web of Science, Google Scholar
17. U. Oertel, Closed incompressible surfaces in complements of star links, Pacific. J. Math. 111 (1984) 209–230. Crossref, Web of Science, Google Scholar
18. N. Owad, Stair-step bridge spectrum does not imply high distance, preprint (2015), arXiv:1510.08469. Google Scholar
19. D. Rolfsen, Knots and Links (Publish or Perish, 1976). Google Scholar
20. H. Schubert, Über eine numerische Knoteninvariante, Math. Z. 61 (1954) 245–288. Crossref, Google Scholar
21. J. Schultens, Additivity of bridge numbers of knots, Math. Proc. Cambridge Philos. Soc. 135 (2003) 539–544. Crossref, Web of Science, Google Scholar
22. M. Tomova, Multiple bridge surfaces restrict knot distance, Algebr. Geom. Topol. 7 (2007) 957–1006. Crossref, Web of Science, Google Scholar
23. F. Waldhausen, Eine Klasse von 3-dimensionalen Mannigfaltigkeiten. I., Invent. Math. 3 (1967) 308–333. Crossref, Google Scholar
24. A. Zupan, Properties of knots preserved by cabling, Comm. Anal. Geom. 19(3) (2011) 541–562. Crossref, Web of Science, Google Scholar
25. A. Zupan, Bridge spectra of iterated torus knots, Comm. Anal. Geom. 22(5) (2014) 931–963. Crossref, Web of Science, Google Scholar