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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



A family of links and the Conway calculus

Author: Cole A. Giller
Journal: Trans. Amer. Math. Soc. 270 (1982), 75-109
MSC: Primary 57M25; Secondary 57M12
MathSciNet review: 642331
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Abstract: In 1969, J. H. Conway gave efficient methods of calculating abelian invariants of classical knots and links. The present paper includes a detailed exposition (with new proofs) of these methods and extensions in several directions.

The main application given here is as follows. A link $ L$ of two unknotted components in $ {S^3}$ has the distinct lifting property for $ p$ if the lifts of each component to the $ p$-fold cover of $ {S^3}$ branched along the other are distinct. The $ p$-fold covers of these lifts are homeomorphic, and so $ L$ gives an example of two distinct knots with the same $ p$-fold cover. The above machinery is then used to construct an infinite family of links, each with the distinct lifting property for all $ p \geqslant 2$.

References [Enhancements On Off] (What's this?)

  • [C1] J. H. Conway, An enumeration of knots and links, and some of their algebraic properties, Computational Problems in Abstract Algebra (John Leech, ed.), Pergamon Press, Oxford and New York, 1969. MR 0258014 (41:2661)
  • [C2] -, Talks at Cambridge Math. Conf. (Summer, 1979).
  • [I] I. M. Isaacs, Character theory of finite groups, Academic Press, New York, 1976. MR 0460423 (57:417)
  • [K] L. Kauffman, The Conway polynomial, Topology 20 (1981), 101-108. MR 592573 (81m:57004)
  • [KT] S. Kinoshita and H. Terasaka, On unions of knots, Osaka Math. J. 9 (1957), 131-153. MR 0098386 (20:4846)
  • [Kb] R. Kirby, Problems in low dimensional manifold theory, Proc. Sympos. Pure Math., vol. 32, part 2, Amer. Math. Soc., Providence, R. I., 1978, pp. 273-312. MR 520548 (80g:57002)
  • [M] J. M. Montesinos, Surgery on links and double branched covers of $ {S^3}$, Knots, Groups and $ 3$-Manifolds (L. P. Neuwirth, ed.), Ann. of Math. Studies, no. 84, Princeton Univ. Press, Princeton, N. J., 1975. MR 0380802 (52:1699)
  • [M1] K. Murasugi, On periodic knots, Comment. Math. Helv. 46 (1971), 162-174. MR 0292060 (45:1148)
  • [M2] -, On the signature of links, Topology 9 (1970), 283-298. MR 0261585 (41:6198)
  • [M3] -, On a certain numerical invariant of link types, Trans. Amer. Math. Soc. 117 (1965), 387-422. MR 0171275 (30:1506)
  • [Ri] R. Riley, Homomorphisms of knot groups on finite groups, Math. Comp. 25 (1971), 603-619. MR 0295332 (45:4399)
  • [R] D. Rolfson, Knots and links, Publish or Perish, Berkeley, Calif., 1976. MR 0515288 (58:24236)
  • [S] H. Schubert, Knoten mit Zwei Brücken, Math. Z. 65 (1956), 133-170. MR 0082104 (18:498e)
  • [V] O. Ya. Viro, Nonprojecting isotopies and knots with homeomorphic coverings, J. Soviet Math. 12 (1979), 86-96.

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Article copyright: © Copyright 1982 American Mathematical Society

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