On the Robertello invariants of proper links

Akio Kawauchi
1984
Robertello's invariant of a classical knot in [9] was generalized by Gordon in [2] to an invariant of a knot in a Z-homology 3-sphere, and by the author in [5] to an invariant, S(kdS), of a knot k in a Z 2 -homology 3-sρhere S. In this paper, we shall generalize this invariant to two mutually related invariants, δ o (Lc*S) and 8(LdS), of a proper link L in a Z 2 -homology 3-sphere S. In the case of a classical proper link, this S 0 -invariant can be considered as an invariant suggested by
more » ... ello in [9, Theorem 2]. A difference between δ o (LcS) and 8(LdS) is that S 0 (LcS) is generally an oriented link type invariant, but δ(LdS) is an unoriented link type invariant. A proper link in a Z 2 -homology 3-sphere (which is not a Z-homology 3-sphere) naturally occurs when considering a branched cyclic covering of a 3-sphere, branched along a certain proper link. (If the number of the components of the link is >2, the branched covering space can not be a Z-homology 3-sphere by the Smith theory.) So, we consider a proper link I in a Z 2 -homology 3-sphere 5, obtained from a proper link L in a Z 2 -homology 3-sphere S by taking a branched cyclic covering, branched along L. When the covering degree is prime, we shall establish a relationship between 8(LdS) and S(LcS) and then a relationship between 8 0 (L d S) and 8 0 (L c S). In Section 1 we define and discuss the slope of a link in a 3-manifold as a generalization of the slope of a knot in a 3-manifold, introduced in [5]. In Section 2 the δ o -invariant and the δ-invariant are defined and studied. Section 3 deals with relationships between δ(LcS) and 8(LdS) and between 8 0 (LdS) andδ o (Lc5). Throughout this paper spaces and maps will be considered in the piecewise linear category, and notations and conventions will be the same as those of [5] unless otherwise stated. The slope of a link in a 3-manifold Let M be a connected oriented 3-manifold. Let L be an oriented link with r components in the interior of M. Let o(L) denote the order (>1) of
doi:10.18910/4202 fatcat:g7d2wwybhvhidcdkuxxxd2ca6i