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GROUPS OF LOCALLY-FLAT DISK KNOTS AND NON-LOCALLY-FLAT SPHERE KNOTS

GREG FRIEDMAN

Department of Mathematics, Yale University, 10 Hillhouse Ave., PO Box 208283, New Haven, CT 06520, USA

https://doi.org/10.1142/S0218216505003786Cited by:3 (Source: Crossref)

Abstract

The classical knot groups are the fundamental groups of the complements of smooth or piecewise-linear (PL) locally-flat knots. For PL knots that are not locally-flat, there is a pair of interesting groups to study: the fundamental group of the knot complement and that of the complement of the "boundary knot" that occurs around the singular set, the set of points at which the embedding is not locally-flat. If a knot has only point singularities, this is equivalent to studying the groups of a PL locally-flat disk knot and its boundary sphere knot; in this case, we obtain a complete classification of all such group pairs in dimension ≥6. For more general knots, we also obtain complete classifications of these group pairs under certain restrictions on the singularities. Finally, we use spinning constructions to realize further examples of boundary knot groups.

Keywords:

AMSC: Primary 57Q45, Primary 57N80, Secondary 57Q35, Secondary 57M05

References

S. Cappell, Superspinning and knot complements, in Topology of Manifolds (Proc. Inst. Univ. of Georgia, Athens, Ga, 1969), (Chicago, Markham, 1970), pp. 358–383 . Google Scholar
S. E. Cappell and J. L. Shaneson, Ann. Math. 134, 325 (1991). Crossref, Web of Science, Google Scholar
A. Borel et al. , Intersection Cohomology , Progress in Mathematics 50 ( Birkhauser , Boston , 1984 ) . Crossref, Google Scholar
G. Friedman, Indiana Univ. Math. J. 52(6), 1479 (2003). Crossref, Web of Science, Google Scholar
G. Friedman, Topology 43, 71 (2004). Crossref, Web of Science, Google Scholar
G. Friedman, Knot spinning, to appear in The Hankbook of Knot Theory, eds. M. Thistlethwaite and W. Menasco . Google Scholar
G. Friedman, Topol. Appl. 134(2), 69 (2003). Crossref, Web of Science, Google Scholar
G. Friedman, Polynomial invariants of non-locally-flat knots, Ph.D. thesis, New York University (2001) . Google Scholar
M. A. Kervaire, Bull. Soc. Math. France 93, 225 (1965). Google Scholar
J. Levine, Ann. Math. 84(2), 537 (1966). Crossref, Web of Science, Google Scholar
J. Levine, Trans. Amer. Math. Soc. 229, 1 (1977). Crossref, Web of Science, Google Scholar
J. Milnor, Conference on the Topology of Manifolds (Boston), ed. J. G. Hocking (PWS Publishing Company, 1968) pp. 115–133. Google Scholar
J. Milnor and R. H. Fox, Osaka J. Math. 3, 257 (1966). Web of Science, Google Scholar
D. Roseman, Topol. Appl. 31, 225 (1989). Crossref, Web of Science, Google Scholar
E. H. Spanier , Algebraic Topology ( Springer-Verlag , New York , 1966 ) . Google Scholar
A. I. Suciu, Comment. Math. Helv. 67, 47 (1992). Crossref, Web of Science, Google Scholar
C. T. C. Wall , Surgery on Compact Manifolds ( Academic Press , London, New York , 1970 ) . Google Scholar
E. C. Zeeman, Trans. Amer. Math. Soc. 115, 471 (1965). Crossref, Web of Science, Google Scholar