Abelian invariants of satellite knots

Abstract

A knot in S 3 whose complement contains an essential** torus is called a satellite knot. In this paper we discuss algebraic invariants of satellite knots, giving short proofs of some known results as well as new results. To each essential torus in the complement of an oriented satellite knot S , one may associate two oriented knots C and E (the companion and embellishment) and an integer w (the winding number). These are defined precisely below. In the late forties, Seifert [S] showed how to compute the Alexander polynomial of S in terms of w and the polynomials of C and E . Implicit in his work is a description of the Alexander module of S . Shinohara [SI,$2] recovered Seifert's results and computed the signature of S using an illuminating description of the infinite cyclic cover M S of S (recalled in ~I below as built up out of the covers of C and E . This description of M S is in essence also due to Seifert ([S] pp. 25, 28). Using it, Kearton [K] stated (without proof) various properties of the Blanchfield palring of S , and deduced a formula for the p-signatures of S (obtained independently by Litherland [L] from a 4-dimensional viewpoint). In §2 we give a complete description of the Blanchfield pairing of S . It depends only on w and the Blanchfield pairings of C and E . (In contrast S cannot be recovered from w , C and E .) In theory one may then compute all the abelian invariants of S from w and the associated invariants of C and E , as the Blanchfield pairing of a knot determines its Seifert form [T2]. This seems difficult in practice however, and so it is appropriate to give more direct computations using the description of M S This is done in §3 for the quadratic form of S , which recovers Shinohara's computation of the signature and gives a formula for the rational Witt invariants of S .