Abstract
In [4] and [5] it was shown that there exist classes of non-fibred knots with algebraically unknotted minimal spanning surfaces, such that the commutator subgroup of the knot group is residually a finite p-group since it can be built up from a free group by adjoining a countable sequence of roots. Here we extend the class of knots, whose commutator subgroups are residually a finite p-group, to include certain knots whose commutator subgroup is the union of absolutely parafree groups which cannot be obtained by adjoining roots to a free group. The extended class now includes all knots in the classical knot table, and it follows that these knots have residually finite groups.
Supported in part by the Canadian National Research Council, Grant #A8207.
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References
R. B. J. T. Allenby and R. J. Gregorac, Residual properties of nilpotent and supersolvable groups, J. Algebra, 23 (1972), 565–573.
G. Baumslag, Groups with the same lower central sequence as a relatively free group, I. II., Trans. Amer. Math. Soc., 129 (1967), 308–321, and 142 (1969), 507–538. MR 36 #248 and 39 #6959.
R. H. Fox, Some problems in knot theory, Topology of 3-Manifolds and Related Topics (Proc. The University of Georgia Institute, 1961), Prentice-Hall, Englewood Cliffs, N. J., 1962.
E. J. Mayland, Jr., On residually finite knot groups, Trans. Amer. Math. Soc. 168 (1972), 221–232.
_____, Two-bridge knots have residually finite groups, to appear: Proc. 2nd World Conference on Group Theory (Canberra, 1973), Springer, Berlin, 1974.
L. Neuwirth, Knot Theory, Princeton Univ. Press, Princeton, N. J., 1965.
K. Reidemeister, Knoten theorie, Chelsea, New York, 1948.
H. Seifert, Uber das Geschlecht von Knoten, Math. Ann., 110 (1934–1935), 571–592.
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© 1975 Springer-Verlag
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Mayland, E.J. (1975). The residual finiteness of the groups of classical knots. In: Glaser, L.C., Rushing, T.B. (eds) Geometric Topology. Lecture Notes in Mathematics, vol 438. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0066129
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DOI: https://doi.org/10.1007/BFb0066129
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