New examples of non-slice, algebraically slice knots

Author:
Charles Livingston

Journal:
Proc. Amer. Math. Soc. **130** (2002), 1551-1555

MSC (1991):
Primary 57M25, 57N70, 57Q60

DOI:
https://doi.org/10.1090/S0002-9939-01-06201-3

Published electronically:
October 12, 2001

MathSciNet review:
1879982

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: For , if the Seifert form of a knotted -sphere in has a metabolizer, then the knot is slice. Casson and Gordon proved that this is false in dimension three. However, in the three-dimensional case it is true that if the metabolizer has a basis represented by a strongly slice link, then is slice. The question has been asked as to whether it is sufficient that each basis element is represented by a slice knot to assure that is slice. For genus one knots this is of course true; here we present genus two counterexamples.

**1.**S. Akbulut and R. Kirby,*Branched covers of surfaces in -manifolds*, Math. Ann. 252, (1979/80), 111-131. MR**82j:57001****2.**G. Burde and H. Zieschang,*Knots*, de Gruyter Studies in Mathematics, 5, Walter de Gruyter & Co., Berlin-New York, 1985. MR**87b:57004****3.**A. Casson and C. Gordon,*Cobordism of classical knots*, A la recherche de la Topologie perdue, ed. by Guillou and Marin, Progress in Mathematics, Volume 62, 1986. (Originally published as Orsay Preprint, 1975.) CMP**19:16****4.**A. Casson and C. Gordon,*On slice knots in dimension three*, Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Part 2, pp. 39-53 Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I., 1978. MR**81g:57003****5.**T. Cochran, K. Orr, and P. Teichner,*Knot concordance, Whitney towers and signatures*, preprint 2000, (ArXiv version at front.math.ucdavis.edu/math.GT/9908117).**6.**M. Freedman and F. Quinn,*Topology of 4-manifolds*, Princeton Mathematical Series, 39, Princeton University Press, Princeton, NJ, 1990. MR**94b:57021****7.**P. Gilmer,*Slice knots in*, Quart. J. Math. Oxford Ser. (2) 34, (1983), no. 135, 305-322. MR**85d:57004****8.**P. Gilmer and C. Livingston,*The Casson-Gordon invariant and link concordance*, Topology 31, (1992), no. 3, 475-492. MR**93h:57037****9.**J. Levine,*Knot cobordism groups in codimension two*, Comment. Math. Helv. 44, (1969), 229-244. MR**39:7618****10.**J. Levine,*Invariants of knot cobordism*, Invent. Math. 8, (1969), 98-110. MR**40:6563****11.**R. Litherland,*A formula for the Casson-Gordon invariant of a knot*, preprint.**12.**R. Litherland,*Cobordism of Satellite Knots*, Four-Manifold Theory, Contemporary Mathematics, eds. C. Gordon and R. Kirby, American Mathematical Society, Providence, R.I., 1984, 327-362. MR**86k:57003****13.**D. Rolfsen,*Knots and Links*, Publish or Perish, Berkeley, CA, 1976. MR**58:24236****14.**A. Tristram,*Some cobordism invariants for links*, Proc. Camb. Phil. Soc. 66, (1969), 251-264. MR**40:2104**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
57M25,
57N70,
57Q60

Retrieve articles in all journals with MSC (1991): 57M25, 57N70, 57Q60

Additional Information

**Charles Livingston**

Affiliation:
Department of Mathematics, Indiana University, Bloomington, Indiana 47405

Email:
livingst@indiana.edu

DOI:
https://doi.org/10.1090/S0002-9939-01-06201-3

Keywords:
Knot concordance,
algebraically slice

Received by editor(s):
August 10, 2000

Received by editor(s) in revised form:
November 10, 2000

Published electronically:
October 12, 2001

Communicated by:
Ronald A. Fintushel

Article copyright:
© Copyright 2001
American Mathematical Society