No Access

LINK HOMOTOPY INVARIANT QUANDLES

JAMES R. HUGHES

Elizabethtown College, Elizabethtown, Pennsylvania 17022, USA

https://doi.org/10.1142/S0218216511008930Cited by:2 (Source: Crossref)

Abstract

We consider several approaches to defining a link homotopy version of the fundamental quandle Q(L) of a link L in S³. We first define the reduced fundamental quandle RQ(L) as a quotient of Q(L). We show that RQ(L) is a link homotopy invariant that carries at least as much information as the meridian-preserving isomorphism class of Milnor's reduced group RG(L). We then show that operator reduction, a plausible alternative approach to defining RQ(L), fails to yield a link homotopy invariant. Finally, we give a geometric characterization of RQ(L), and offer a caveat regarding a seemingly simpler approach.

Keywords:

AMSC: 57M25

References

R. Fenn and C. Rourke, J. Knot Theory Ramifications 1, 343 (1992). Link, Google Scholar
N. Habegger and X. S. Lin, J. Amer. Math. Soc. 3, 389 (1990), DOI: 10.1090/S0894-0347-1990-1026062-0. Crossref, Web of Science, Google Scholar
J. Hempel, 3-Manifolds, Annals of Mathematics Studies, Vol.86 (1976) . Google Scholar
J. Hughes, J. Knot Theory Ramifications 2, 37 (1993). Link, Web of Science, Google Scholar
J. Hughes, J. Knot Theory Ramifications 7, 925 (1998), DOI: 10.1142/S0218216598000498. Link, Web of Science, Google Scholar
J. Hughes, J. Knot Theory Ramifications 12, 375 (2003). Link, Web of Science, Google Scholar
J. Hughes, Topol. Appl. 148, 55 (2005). Crossref, Web of Science, Google Scholar
D. Joyce, J. Pure Appl. Algebra 23, 37 (1982), DOI: 10.1016/0022-4049(82)90077-9. Crossref, Web of Science, Google Scholar
S. Kamada, Geometry and Topology Monographs 4, 103 (2002). Google Scholar
L. Kaufmann, Knot crystals: Classical knot theory in modern guise, in Knots and Physics . Google Scholar
J. Levine, Trans. Amer. Math. Soc. 306, 361 (1988), DOI: 10.1090/S0002-9947-1988-0927695-7. Crossref, Web of Science, Google Scholar
X. S. Lin, L'Enseignement Math. 47, 315 (2001). Web of Science, Google Scholar
B. Mellor and D. Thurston, J. Knot Theory Ramifications 10, 1025 (2001). Link, Web of Science, Google Scholar
J. Milnor, Ann. Math. 59, 177 (1954), DOI: 10.2307/1969685. Crossref, Web of Science, Google Scholar