On dihedral covering spaces of knots

Abstract

With an appropriate choice of notation, the combinatorial methods of [13] may be conveniently applied to the irregular coverings studied by Reidemeister and Bankwitz [16; 1], revealing simple formulas for, and new relationships among, linking numbers between branch curves 1. To each homomorphism of the group of a tame knot onto an abstract group fr of finite order ~ there corresponds a partially ordered set of covering spaces of S 3, branched along the knot. Each distinct/-sheeted element may be associated biuniquely with a class of conjugate subgroups of index i in fr and constructed explicitly by following the homomorphism with the associated coset representation of ff onto a transitive permutation group of degree i. Cf. [13; 3, Ch. XII]. Where one such class is "contained in" another (in the sense of inclusion for some choice of particular subgroups) the associated covering covers the other by a covering map which may be thought of as the identification of corresponding points in various copies of S 3. The set of such coverings and covering maps forms a commutative diagram. Cf. [14, p. 49]. Since the w-sheeted covering which covers all others is regular, its covering translations restrict the linking numbers between pairs of its branch curves; for if one such pair may be moved to another, the linking numbers between them are obviously equal. Linking numbers in this maximal covering may then be trickled down to the lesser-sheeted ones by a simple transfer argument, formulated in the following generality by the late R.H.Fox. Let k be a link in a closed 3-manifold M with components kl, . . . ,k , and let f : M ~ M be a covering of finite de_gree branched over k. Let kil . . . . . kim,) be the components of k / = f l ( k 3 in M. Let ci, be the branching index o f f at k/a and let d/a be the degree of the mapping f[kia. The links k~ and k~ having been so oriented that d~,>0, let kl and k~a denote also the 1-cycles carried by these oriented knots. If k~ is a torsion cycle, then there is a 2-chain C~ in M whose boundary OC~ is, say, tik i. If kj is also a torsion cycle, then the linking number L(ki, k) is defined and_equal to (1/t~)S(Ci, k)where S denotes the intersection number. The transfer C~ of C~ is a 2-chain in M that is easily seen to have the