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FIBERED TORTI-RATIONAL KNOTS

MIKAMI HIRASAWA

Department of Mathematics, Nagoya Institute of Technology, Nagoya Aichi 466-8555, Japan

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KUNIO MURASUGI

Department of Mathematics, University of Toronto, ON M5S2E4, Canada

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https://doi.org/10.1142/S0218216510008431Cited by:2 (Source: Crossref)

Abstract

A torti-rational knot, denoted by K(2α, β|r), is a knot obtained from the 2-bridge link B(2α, β) by applying Dehn twists an arbitrary number of times, r, along one component of B(2α, β). We determine the genus of K(2α, β|r) and solve a question of when K(2α, β|r) is fibered. In most cases, the Alexander polynomials determine the genus and fiberedness of these knots. We develop both algebraic and geometric techniques to describe the genus and fiberedness by means of continued fraction expansions of β/2α. Then, we explicitly construct minimal genus Seifert surfaces. As an application, we solve the same question for the satellite knots of tunnel number one.

Dedicated to Professor Akio Kawauchi for his 60th Birthday

Keywords:

AMSC: 57M25, 57M27

References

M. Aït-Nouh, D. Matignon and K. Motegi, Ann. Math. Blaise Pascal 13, 31 (2006). Crossref, Google Scholar
K. S. Brown, Invent. Math 90, 479 (1987), DOI: 10.1007/BF01389176. Crossref, Web of Science, Google Scholar
G. Burde and H. Zieschang , Knots , 2nd edn. , de Gruyter Studies in Mathematics 5 ( Walter de Gruyter , 2003 ) . Google Scholar
D. Eisenbud and W. Neumann , Three-Dimensional Link Theory and Invariants of Plane Curve Singularities , Annals of Mathematics Studies 110 ( Princeton University Press , Princeton, NJ , 1985 ) . Google Scholar
D. Gabai, Genera of the Arborescent Links, Memoirs of the American Mathematical Society 59 (1986) pp. 1–98. Google Scholar
D. Gabai, Comment. Math. Helv. 61, 519 (1986), DOI: 10.1007/BF02621931. Crossref, Web of Science, Google Scholar
H. Goda, C. Hayashi and H.-J. Song, Dehn surgeries on 2-bridge links which yield reducible 3-manifolds, preprint (2005) , arXiv:math.GT/0512116 . Google Scholar
H. Goda and M. Teragaito, Tokyo J. Math. 22, 99 (1999), DOI: 10.3836/tjm/1270041615. Crossref, Web of Science, Google Scholar
F. González-Acuña, Bol. Soc. Mat Mexicana (2) 15, 58 (1970). Google Scholar
M. Hirasawa and K. Murasugi, Osaka J. Math. 44, 11 (2007). Web of Science, Google Scholar
M. Hirasawa, K. Murasugi and D. Silver, Hiroshima Math. J. 38, 411 (2008). Crossref, Web of Science, Google Scholar
T. Kanenobu, J. Austral. Math. Soc. Ser. A 36, 59 (1984), DOI: 10.1017/S1446788700027336. Crossref, Google Scholar
T. Kanenobu and Y. Miyazawa, Kobe J. Math. 9, 171 (1992). Web of Science, Google Scholar
M. Kidwell, Math. Sem. Notes Kobe Univ. 10, 387 (1982). Google Scholar
M. Kouno, K. Motegi and T. Shibuya, J. Math. Soc. Japan 44, 199 (1992), DOI: 10.2969/jmsj/04420199. Crossref, Web of Science, Google Scholar
T. Maeda and K. Murasugi, Contemp. Math. 20, 271 (1983). Crossref, Google Scholar
K. Morimoto and M. Sakuma, Math. Ann. 289, 143 (1991), DOI: 10.1007/BF01446565. Crossref, Web of Science, Google Scholar
M. Scharlemann, Trans. Amer. Math. Soc. 356(4), 1385 (2004), DOI: 10.1090/S0002-9947-03-03182-9. Crossref, Web of Science, Google Scholar
G. Torres, Ann. Math. (2) 57, 57 (1953), DOI: 10.2307/1969726. Crossref, Web of Science, Google Scholar