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

DOI:
https://doi.org/10.1090/S0002-9947-1982-0642331-X

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 of two unknotted components in has the *distinct lifting property* for if the lifts of each component to the -fold cover of branched along the other are distinct. The -fold covers of these lifts are homeomorphic, and so gives an example of two distinct knots with the same -fold cover. The above machinery is then used to construct an infinite family of links, each with the distinct lifting property for all .

**[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*, Knots, Groups and -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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DOI:
https://doi.org/10.1090/S0002-9947-1982-0642331-X

Article copyright:
© Copyright 1982
American Mathematical Society