Fibered 2-knots and lens spaces

Abstract

and pi>qi>0 (cf. [10], [13]). The punctured manifold M° means the space obtained from M by removing an open 3-ball. We consider fibered 2-knoΐs in S with fiber M°, which we call M°-fibered 2-knots. For example, a connected sum of 2-twist-spun knots of 2-bridge knots is such a fibered 2-knot. We shall determine possible M°-fibered 2-knots in S (or more generally, a homology 4-sρhere Σ) for all M. There are two reasons for selecting these fibered 2-knots: (1) Any 2-knot with Seifert manifold a connected sum of lens spaces is determined by its exterior [3], [4]. In particular, any fibered 2-knot with fiber a connected sum of lens spaces is determined by its monodromy (precisely, the class of its monodromy in the diίfeotopy group of the fiber). (2) The diffeotopy groups of all lens spaces are computed by Bonahon [1] and Hodgson-Rubinstein [7]. In section 3, we prove that any fibered 2-knot with fiber a punctured lens space L(p, q)° is the 2-twist-spun knot of K(p} q). In general, we show in section 4 that any M°-fibered 2-knot with cyclic monodromy is a cable knot about the 2-twist-spun knot of a 2-bridge knot. In section 5, we consider branched covering spaces of cable knots, and observe, for each odd r>l, the existence of a 2-knot which is the same fixed point set of many inequivalent semi-free ^-actions on S (cf. [2], [5], [14]). I would like to express my gratitude to Professor Akio Kawauchi for suggesting the problem, Professor Fujitsugu Hosokawa and Professor Yasutaka Nakanishi for leading me throughout.