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ALEXANDER POLYNOMIALS OF ALTERNATING KNOTS OF GENUS TWO II

IN DAE JONG

Osaka City University, Advanced Mathematical Institute, Osaka 558-8585, Japan

https://doi.org/10.1142/S0218216510008285Cited by:12 (Source: Crossref)

Abstract

We give a family of linear inequalities which strictly estimate relation among the coefficients of the Alexander polynomials of alternating knots of genus two. We also give such families for positive knots of genus two, and for homogeneous knots of genus two. As an application, we determine the alternating knots of genus two such that the leading coefficients of them are less than or equal to three.

Keywords:

AMSC: 57M25, 57M27

References

T. Abe, J. Knot Theory Ramifications 18, 363 (2009), DOI: 10.1142/S021821650900694X. Link, Web of Science, Google Scholar
T. Abe and K. Kishimoto, J. Knot Theory Ramifications (2010). Google Scholar
J. W. Alexander, Trans. Amer. Math. Soc. 30, 275 (1928). Crossref, Web of Science, Google Scholar
J. H. Conway, Computational Problems in Abstract Algebra (1970) pp. 329–358. Crossref, Google Scholar
P. R. Cromwell, J. London Math. Soc. (2) 39, 535 (1989). Web of Science, Google Scholar
P. R. Cromwell , Knots and Links ( Cambridge University Press , 2004 ) . Crossref, Google Scholar
R. H. Crowell, Ann. Math. 69(2), 258 (1959), DOI: 10.2307/1970181. Crossref, Web of Science, Google Scholar
D. De Wit and J. Links, J. Knot Theory Ramifications 16(8), 1021 (2007), DOI: 10.1142/S0218216507005658. Link, Web of Science, Google Scholar
R. H. Fox, Topology of 3-Manifolds and Related Topics (1962) pp. 168–176. Google Scholar
R. I. Hartley, J. Austral. Math. Soc. Ser. A 28, 241 (1979), DOI: 10.1017/S1446788700015743. Crossref, Google Scholar
K. Ichihara, I. D. Jong and S. Mizushima, Seifert fibered surgeries on alternating Montesinos knots, in preparation . Google Scholar
I. D. Jong, Osaka J. Math. 46, 353 (2009). Web of Science, Google Scholar
T. Kanenobu, Math. Ann. 275, 555 (1986), DOI: 10.1007/BF01459137. Crossref, Web of Science, Google Scholar
T. Kanenobu, Proc. Amer. Math. Soc. 97(1), 158 (1986). Crossref, Web of Science, Google Scholar
T. Kanenobu, Topol. Appl. 157(1), 302 (2010), DOI: 10.1016/j.topol.2009.04.025. Crossref, Web of Science, Google Scholar
A. Kawauchi , A Survey of Knot Theory ( Birckhäuser-Verlag , Basel , 1996 ) . Google Scholar
A. Kawauchi, Topol. Appl. 157, 274 (2010), DOI: 10.1016/j.topol.2009.04.027. Crossref, Web of Science, Google Scholar
S. Kinoshita and H. Terasaka, Osaka Math. J. 9, 131 (1957). Web of Science, Google Scholar
C. Lamm, Osaka J. Math. 37(3), 537 (2000). Web of Science, Google Scholar
E. S. Lee, Adv. Math. 197(2), 554 (2005), DOI: 10.1016/j.aim.2004.10.015. Crossref, Web of Science, Google Scholar
K. Murasugi, J. Math. Soc. Japan 10, 94 (1958), DOI: 10.2969/jmsj/01010094. Crossref, Web of Science, Google Scholar
K. Murasugi, J. Math. Soc. Japan 10, 235 (1958), DOI: 10.2969/jmsj/01030235. Crossref, Web of Science, Google Scholar
K. Murasugi, Trans. Amer. Math. Soc. 117, 387 (1965). Crossref, Web of Science, Google Scholar
K. Murasugi, J. Austral. Math. Soc. Ser. A 39(3), 317 (1985), DOI: 10.1017/S1446788700026094. Crossref, Google Scholar
K. Murasugi , Knot Theory and Its Applications ( Birckhäuser-Verlag , Boston , 1996 ) . Google Scholar
P. Ozsváth and Z. Szabó, Geom. Topol. 7, 225 (2003). Crossref, Web of Science, Google Scholar
D. Rolfsen , Knots and Links ( AMS Chelsea publishing , 2003 ) . Crossref, Google Scholar
H. Seifert, Math. Ann. 110, 571 (1935), DOI: 10.1007/BF01448044. Crossref, Google Scholar
A. Stoimenow, Fund. Math. 200, 1 (2008), DOI: 10.4064/fm200-1-1. Crossref, Web of Science, Google Scholar
J. M. van Buskirk, Knot Theory and Manifolds (Vancouver, 1983), Lecture Notes in Mathematics 1144 (Springer, Berlin, 1985) pp. 146–159. Crossref, Google Scholar