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Some invariants of Pretzel links

Published online by Cambridge University Press:  17 April 2009

Dongseok Kim  and
Jaeun Lee
Dongseok Kim
Affiliation:
Department of Mathematics, Yeungnam University, Kyongsan, 712–749, Korea, e-mail: [email protected], [email protected]
Jaeun Lee
Affiliation:
Department of Mathematics, Yeungnam University, Kyongsan, 712–749, Korea, e-mail: [email protected], [email protected]
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We show that nontrivial classical pretzel knots L (p, q, r) are hyperbolic with eight exceptions which are torus knots. We find Conway polynomials of n-pretzel links using a new computation tree. As applications, we compute the genera of n-pretzel links using these polynomials and find the basket number of pretzel links by showing that the genus and the canonical genus of a pretzel link are the same.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 75 , Issue 2 , April 2007 , pp. 253 - 271
Copyright
Copyright © Australian Mathematical Society 2007

References

[1]Adams, C., ‘Almost alternating links’, Topology Appl. 46 (1992), 151–165.Google Scholar
[2]Adams, C., The knot book (W.H. Freeman and Company, New York 1994).Google Scholar
[3]Bae, Y., Kim, D. and Park, C., ‘Basket, flat plumbing and flat plumbing basket numbers of links’, (preprint, arXiv:GT/0607079).Google Scholar
[4]Bedient, R., ‘Double branched covers and pretzel knots’, Pacific J. Math 112 (1984), 265–272.CrossRefGoogle Scholar
[5]Brittenham, M., ‘Free genus one knots with large volume’, Pacific J. Math. 201 (2001), 61–82.Google Scholar
[6]Cimasoni, D., ‘A geometric construction of the Conway potential function’, Comment. Math. Helv. 79 (2004), 126–146.Google Scholar
[7]Conway, J., ‘An enumeration of knots and links, and some of their algebraic properties’, in Computational problems in abstract algebra (Pergamon Press, Oxford, 1970), pp. 329–358.Google Scholar
[8]Crowell, R., ‘Genus of alternating link types’, Ann. of Math. 69 (1959), 258–275.Google Scholar
[9]Franks, J. and Williams, R., ‘Braids and the Jones polynomials’, Trans. Amer. Math. Soc. 303 (1987), 97–108.Google Scholar
[10]Gabai, D., ‘The Murasugi sum is a natural geometric operation’, in Low-Dimensional Topology (San Francisco, CA, USA, 1981) (Amer. Math. Soc., Providence, RI, 1983), pp. 131–143.Google Scholar
[11]Gabai, D., ‘The Murasugi sum is a natural geometric operation II’, in Combinatorial Methods in Topology and Algebraic Geometry (Rochester, NY, USA, 1982) (Amer. Math. Soc., Providence, RI, 1985), pp. 93–100.Google Scholar
[12]Gabai, D., ‘Genera of the arborescent links’, Mem. Amer. Math. Soc. 59 (1986), 1–98.Google Scholar
[13]Hara, M., ‘Q-polynomial of pretzel links’, Tokyo J. Math. 16 (1993), 183–190.Google Scholar
[14]Hare, J., ‘How to construct all fibered knots and links’, Topology 21 (1982), 263–280.Google Scholar
[15]Hara, M., Nakagawa, Y. and Ohyama, Y., ‘The Conway potential functions for pretzel links and Montesinos links’, Kobe J. Math. 6 (1989), 1–21.Google Scholar
[16]Hayashi, C. and Wada, M., ‘Constructing links by plumbing flat annuli’, J. Knot Theory Ramifications 2 (1993), 427–429.Google Scholar
[17]Hoste, J., Thistlethwaite, M. and Weeks, J., ‘The first 1701936 knots’, Math. Intellegencer 20 (1998), 33–48.Google Scholar
[18]Jones, V., ‘Hecke algebra representations of braid groups and link polynomials’, Ann. of Math. 126 (1987), 335–388.CrossRefGoogle Scholar
[19]Hosokawa, F. and Nakanishi, Y., ‘On 3-fold irregular branched covering spaces of pretzel knots’, Osaka J. Math. 23 (1986), 249–254.Google Scholar
[20]Kauffman, L., ‘Combinatorics and knot theory’, Contemp. Math. 20 (1983), 181–200.Google Scholar
[21]Kobayashi, M. and Kobayashi, T., ‘On canonical genus and free genus of knot’, J. Knot Theory Ramifications 5 (1996), 77–85.CrossRefGoogle Scholar
[22]Landvoy, R., ‘The Jones polynomial of pretzel knots and links’, Topology Appl. 83 (1998), 135–147.Google Scholar
[23]Lickorish, W., ‘Prime knots and tangles’, Trans. Amer. Math. Soc. 267 (1981), 321–332.Google Scholar
[24]Menasco, W., ‘Closed incompressible surfaces in alternating knot and link complements’, Topology 23 (1984), 37–44.Google Scholar
[25]Moriah, Y., ‘On the free genus of knots’, Proc. Amer. Math. Soc. (1987), 373–379.CrossRefGoogle Scholar
[26]Morton, H., ‘Fibred knots with a given Alexander polynomial’, in Knots, Braids and Singularities (Plans-sur-Bex, 1982) (Enseign. Math., Geneva, 1983), pp. 205–222.Google Scholar
[27]Murasugi, K., ‘On the genus of the alternating knot, I. II’, J. Math. Soc. Japan 10 (1958), 94–105, 235–248.Google Scholar
[28]Nakagawa, Y., ‘On the Alexander polynomials of pretzel links L (p 1, …,pn)’, Kobe J. Math. 3 (1987), 167–177.Google Scholar
[29]Nakamura, T., ‘On canonical genus of fibered knot’, J. Knot Theory Ramifications 11 (2002), 341–352.CrossRefGoogle Scholar
[30]Rudolph, L., ‘Quasipositive annuli (Constructions of quasipositive knots and links IV.)’, J. Knot Theory Ramifications 1 (1992), 451–466.Google Scholar
[31]Rudolph, L., ‘Hopf plumbing, arborescent Seifert surfaces, baskets, espaliers, and homogeneous braids’, Topology Appl. 116 (2001), 255–277.Google Scholar
[32]Sakuma, M., ‘Minimal genus Seifert surfaces for special arborescent links’, Osaka J. Math. 31 (1994), 861–905.Google Scholar
[33]Seifert, H., ‘Uber das Geschlecht von Knoten’, Math. Ann. 110 (1934), 571–592.Google Scholar
[34]Shinohara, Y., ‘On the signature of pretzel links’, in Topology and computer science(Atami, 1986) (Kinokuniya, Tokyo, 1987), pp. 217–224.Google Scholar
[35]Stallings, J., ‘Constructions of fibred knots and links’, in Algebraic and Geometric Topology, (Proc. Sympos. Pure Math., Stanford Univ., Stanford, CA, 1976), Part 2 (Amer. Math. Soc, Providence, RI, 1978), pp. 55–60.Google Scholar
[36]Torres, G., ‘On the Alexander polynomail’, Ann. of Math. 57 (1953), 57–89.Google Scholar