HNN bases and high-dimensional knots

Author:
Daniel S. Silver

Journal:
Proc. Amer. Math. Soc. **124** (1996), 1247-1252

MSC (1991):
Primary 57Q45; Secondary 20E06, 20F05

DOI:
https://doi.org/10.1090/S0002-9939-96-03520-4

MathSciNet review:
1343725

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: There exists a -knot group having HNN bases of two types: bases that are arbitrarily large finitely presented and bases that are arbitrarily large finitely generated but not finitely presented. Any -knot with such a group has a Seifert manifold that can be converted to a minimal one by a finite sequence of ambient - and -surgeries, but cannot be converted by -surgeries alone.

**1.**G. Baumslag,*A remark on generalized free products*, Proc.Amer. Math. Soc.**13**(1962), 53--54. MR**26:202****2.**G. Baumslag, P.B. Shalen,*Amalgamated products and finitely presented groups*, Comment. Math. Helv.**65**(1990), 243--254. MR**91j:20071****3.**R. Bieri, R. Strebel,*Almost finitely presented soluble groups*, Comment. Math. Helv.**53**(1978), 258--278. MR**58:16890****4.**I.M. Chiswell,*Exact sequences associated with a graph of groups*, J. Pure Appl. Alg.**8**(1976), 63--74. MR**53:3147****5.**C.McA. Gordon,*Homology of groups of surfaces in the 4-sphere*, Math. Proc. Camb. Phil. Soc.**89**(1981), 113--117. MR**83d:57016****6.**J. Hillman,*High dimensional knot groups which are not two-knot groups*, Bull. Austral. Math. Soc.**16**(1977), 449--462. MR**58:31098****7.**M. Kervaire,*On higher dimensional knots*, Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse) (S.S. Cairns, ed.), Princeton University Press, Princeton, 1965, pp. 105--109. MR**31:2732****8.**J. Levine,*Unknotting spheres in codimension two*, Topology**4**(1965), 9--16. MR**31:4045****9.**T. Maeda,*Knotted surfaces in the -sphere with no minimal Seifert manifolds*, preprint.**10.**D.I. Moldavanskii,*Certain subgroups of groups with one defining relation (in Russian)*,

Sibirsk. Math. Z.**8**(1967), 1370--1384. MR**36:3862****11.**B.H. Neumann,*Some remarks on infinite groups*, J. London Math. Soc.**12**(1937), 4--11.**12.**E.S. Rapaport,*Knot-like groups*, Annals of Math. Studies, vol. 84, Princeton Univ. Press, Princeton, 1975, pp. 119--133.**13.**D.S. Silver,*Examples of -knots with no minimal Seifert manifolds*, Math. Proc. Camb. Phil. Soc.**110**(1991), 417--420. MR**92f:57030****14.**D.S. Silver,*On the existence of minimal Seifert manifolds*, Math. Proc. Camb. Phil. Soc.**114**(1993), 103--109. MR**94c:47039****15.**D.S. Silver,*On knot-like groups and ribbon concordance*, J. Pure Appl. Alg.**82**(1992 99--105). MR**94a:57021****16.**J. Stallings,*A finitely presented group whose 3-dimensional integral homology is not finitely generated*, Amer. Journal of Math.**95**(1963), 541--543. MR**28:2139**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
57Q45,
20E06,
20F05

Retrieve articles in all journals with MSC (1991): 57Q45, 20E06, 20F05

Additional Information

**Daniel S. Silver**

Affiliation:
Department of Mathematics and Statistics, University of South Alabama, Mobile, Alabama 36688

Email:
silver@mathstat.usouthal.edu

DOI:
https://doi.org/10.1090/S0002-9939-96-03520-4

Received by editor(s):
May 17, 1994

Communicated by:
James West

Article copyright:
© Copyright 1996
American Mathematical Society