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Alexander polynomials of equivariant slice and ribbon knots in $S^3$

Author(s): James F. Davis; Swatee Naik
Journal: Trans. Amer. Math. Soc. 358 (2006), 2949-2964.
MSC (2000): Primary 57M25
Posted: May 26, 2005
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Abstract | References | Similar articles | Additional information

Abstract: This paper gives an algebraic characterization of Alexander polynomials of equivariant ribbon knots and a factorization condition satisfied by Alexander polynomials of equivariant slice knots.


References:

1.
I. Aitchison and D. Silver, On certain fibred ribbon disc pairs, Trans. Amer. Math. Soc. 306 (1988) 529-550. MR 0933305 (89f:57004)

2.
G. Burde and H. Zieschang, Knots, Studies in Math. 5, de Gruyter, 1985.MR 0808776 (87b:57004)

3.
J. C. Cha and K. H. Ko, On equivariant slice knots, Proc. Amer. Math. Soc. 127 (1999) 2175-2182. MR 1605928 (2000a:57006)

4.
J. C. Cha, A characterization of the Murasugi polynomial of an equivariant slice knot, (2004) preprint.

5.
J. Conway, An enumeration of knots and links and some of their algebraic properties in Computational Problems in Abstract Algebra, Proc. Conf. Oxford 1967, 329-358 (ed. John Leech) Pergamon Press, 1970. MR 0258014 (41:2661)

6.
J. F. Davis and C. Livingston, Alexander polynomials of periodic knots, Topology 30 (1991) 551-564. MR 1133872 (92k:57008)

7.
C. H. Giffen, The generalized Smith conjecture, Amer. J. Math. 88 (1966) 187-198. MR 0198462 (33:6620)

8.
R. H. Fox, A quick trip through knot theory, in Topology of 3-Manifolds and Related Topics (Proc. The Univ. of Georgia Institute, 1961), 120-167, (ed. M.K. Fort) Prentice Hall, 1962. MR 0140099 (25:3522)

9.
R. H. Fox, Some problems in knot theory, in Topology of 3-Manifolds and Related Topics (Proc. The Univ. of Georgia Institute, 1961), 168-176, (ed. M.K. Fort) Prentice Hall, 1962. MR 0140100 (25:3523)

10.
R. H. Fox and J. W. Milnor, Singularities of 2-spheres in 4-space and cobordism of knots, Osaka J. Math. 3 (1966) 257-267. MR 0211392 (35:2273)

11.
R. E. Gompf and A. I. Stipsicz, 4-manifolds and Kirby Calculus, Amer. Math. Soc., 1999. MR 1707327 (2000h:57038)

12.
J. A. Hillman, Polynomials of equivariantly slice knots, (2003) preprint.

13.
R. Kirby, Problems in low dimensional topology, in Geometric Topology (Athens, GA, 1993), 35-473, ed. R. Kirby, Amer. Math. Soc., 1997. MR 1470751

14.
A. A. Kosinski, Differential Manifolds, Academic Press, 1993.MR 1190010 (95b:57001)

15.
K. H. Ko, D. H. Choi, and W. T. Song, Seifert matrices and equivariant concordances of periodic knots, (1997) preprint.

16.
J. Levine, A characterization of knot polynomials, Topology 4 (1965) 135-141. MR 0180964 (31:5194)

17.
J. Milnor, A duality theorem for Reidemeister torsion, Ann. of Math. (2) 76 (1962) 137-147. MR 0141115 (25:4526)

18.
J. Milnor, Whitehead torsion, Bull. Amer. Math. Soc. 72 (1966) 358-426. MR 0196736 (33:4922)

19.
J. Morgan and H. Bass, ed., The Smith Conjecture, Academic Press, 1984. MR 0758459 (86i:57002)

20.
K. Murasugi, On periodic knots, Comment. Math. Helv. 46 (1971) 162-174.MR 0292060 (45:1148)

21.
S. Naik, Equivariant slice knots in $S^3$, Knots `96 (Tokyo), 81-89, Proceedings of the Fifth International Research Institute of Mathematical Society of Japan, held at Waseda University, Tokyo, July 22-26, 1996, ed. S. Suzuki, World Sci. Publishing, 1997.MR 1664952 (99k:57028)

22.
D. S. Rim, Modules over finite groups, Ann. of Math. (2) 69 (1959) 700-712.MR 0104721 (21:3474)

23.
D. Rolfsen, Knots and Links, Mathematics Lecture Series, No. 7. Publish or Perish, Inc., 1976.MR 0515288 (58:24236)

24.
P. A. Smith, Transformations of finite period, Ann. of Math. (2) 39 (1938) 127-164. MR 1503393

25.
J. R. Smith, Acyclic localizations, J. Pure Appl. Algebra 12 (1978) 117-127. MR 0491895 (58:11075)

26.
H. Terasaka On null equivalent knots, Osaka Math. J. 11 (1959) 95-113. MR 0117736 (22:8511)

27.
V. G. Turaev, Reidemeister torsion in knot theory, Russian Math. Surveys 41 (1986) 119-182. MR 0832411 (87i:57009)

28.
C. T. C. Wall, Surgery on Compact Manifolds, Academic Press, 1970.MR 0431216 (55:4217)

29.
G. W. Whitehead, Elements of Homotopy Theory, Springer, 1978.MR 0516508 (80b:55001)


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Additional Information:

James F. Davis
Affiliation: Department of Mathematics, Indiana University, Bloomington, Indiana 47405-4301

Swatee Naik
Affiliation: Department of Mathematics & Statistics, University of Nevada, Reno, Nevada 89557

DOI: 10.1090/S0002-9947-05-03741-4
PII: S 0002-9947(05)03741-4
Received by editor(s): May 20, 2002
Received by editor(s) in revised form: May 11, 2004
Posted: May 26, 2005
Additional Notes: The first author was partially supported by a grant from the National Science Foundation
Copyright of article: Copyright 2005, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.


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