Alexander polynomials of doubly primitive knots

Authors:
Kazuhiro Ichihara, Toshio Saito and Masakazu Teragaito

Journal:
Proc. Amer. Math. Soc. **135** (2007), 605-615

MSC (2000):
Primary 57M25

Published electronically:
August 10, 2006

MathSciNet review:
2255308

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We give a formula for Alexander polynomials of doubly primitive knots. This also gives a practical algorithm to determine the genus of any doubly primitive knot.

**1.**J. Berge,*Some knots with surgeries yielding lens spaces*, unpublished manuscript.**2.**Richard H. Crowell and Ralph H. Fox,*Introduction to knot theory*, Based upon lectures given at Haverford College under the Philips Lecture Program, Ginn and Co., Boston, Mass., 1963. MR**0146828****3.**Marc Culler, C. McA. Gordon, J. Luecke, and Peter B. Shalen,*Dehn surgery on knots*, Ann. of Math. (2)**125**(1987), no. 2, 237–300. MR**881270**, 10.2307/1971311**4.**Ronald Fintushel and Ronald J. Stern,*Constructing lens spaces by surgery on knots*, Math. Z.**175**(1980), no. 1, 33–51. MR**595630**, 10.1007/BF01161380**5.**C. McA. Gordon,*Dehn filling: a survey*, Knot theory (Warsaw, 1995) Banach Center Publ., vol. 42, Polish Acad. Sci. Inst. Math., Warsaw, 1998, pp. 129–144. MR**1634453****6.**T. Kadokami,*Reidemeister torsion of homology lens spaces*, preprint.**7.**T. Kadokami and Y. Yamada,*A deformation of the Alexander polynomials of knots yielding lens spaces*, preprint.**8.**T. Kadokami and Y. Yamada,*Reidemeister torsion and lens surgeries on -pretzel knots*, preprint.**9.**Rob Kirby (ed.),*Problems in low-dimensional topology*, Geometric topology (Athens, GA, 1993) AMS/IP Stud. Adv. Math., vol. 2, Amer. Math. Soc., Providence, RI, 1997, pp. 35–473. MR**1470751****10.**P. Kronheimer, T. Mrowka, P. Ozsváth and Z. Szabó,*Monopoles and lens space surgeries*, preprint,`arXiv:math.GT/0310164`.**11.**Louise Moser,*Elementary surgery along a torus knot*, Pacific J. Math.**38**(1971), 737–745. MR**0383406****12.**Peter Ozsváth and Zoltán Szabó,*On knot Floer homology and lens space surgeries*, Topology**44**(2005), no. 6, 1281–1300. MR**2168576**, 10.1016/j.top.2005.05.001**13.**Dale Rolfsen,*Knots and links*, Publish or Perish, Inc., Berkeley, Calif., 1976. Mathematics Lecture Series, No. 7. MR**0515288**

Dale Rolfsen,*Knots and links*, Mathematics Lecture Series, vol. 7, Publish or Perish, Inc., Houston, TX, 1990. Corrected reprint of the 1976 original. MR**1277811****14.**T. Saito,*Dehn surgery and -knots in lens spaces*, preprint.**15.**T. Saito,*The dual knots of doubly primitive knots*, preprint.

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
57M25

Retrieve articles in all journals with MSC (2000): 57M25

Additional Information

**Kazuhiro Ichihara**

Affiliation:
College of General Education, Osaka Sangyo University, Nakagaito 3-1-1, Daito, Osaka 574-8530, Japan

Email:
ichihara@las.osaka-sandai.ac.jp

**Toshio Saito**

Affiliation:
Graduate School of Humanities and Sciences, Nara Women’s University, Kitauoyanishi-machi, Nara 630-8506, Japan

Email:
tsaito@cc.nara-wu.ac.jp

**Masakazu Teragaito**

Affiliation:
Department of Mathematics and Mathematics Education, Hiroshima University, Kagamiyama 1-1-1, Higashi-hiroshima, Japan 739-8524.

Email:
teragai@hiroshima-u.ac.jp

DOI:
http://dx.doi.org/10.1090/S0002-9939-06-08496-6

Keywords:
Doubly primitive knot,
Alexander polynomial

Received by editor(s):
June 21, 2005

Received by editor(s) in revised form:
September 13, 2005

Published electronically:
August 10, 2006

Additional Notes:
The second author was supported by the 21st Century COE program \lq\lq Towards a New Basic Science; Depth and Synthesis\rq\rq, Osaka University.

The third author was partially supported by the Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (C), 16540071.

Communicated by:
Ronald A. Fintushel

Article copyright:
© Copyright 2006
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.