A family of links and the Conway calculus

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 L of two unknotted components in S 3 {S^3} has the distinct lifting property for p p if the lifts of each component to the p p -fold cover of S 3 {S^3} branched along the other are distinct. The p p -fold covers of these lifts are homeomorphic, and so L L gives an example of two distinct knots with the same p 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 ⩾ 2 p \geqslant 2 .