On the modified Goeritz matrices of 2-periodic links

Abstract

An oriented link ί — k U U kμ of μ components in S3 is called a Ί-periodic link if there is a Z2-action on the pair (S3,^) such that the fixed point set / of the action is homeomorphic to a 1-sphere in S 3 disjoint from ί. It is known that / is unknotted. Hence the quotient map p : S3 -> 53/Z2 is an 2-fold cyclic branched covering branched over p(f) — /* and p(ί) — ί* is also an oriented link in the orbit space 53/Z2 = S3, which is called the factor link of L In this paper, we express a relationship between the modified Goeritz matrices of a 2-periodic link ί and those of its factor link £* and the link ί* U /*. As an application, we give an alternative proof of the Gordon and Litherland's formular([3]): σ(ί)-Lk(ί,f] = σ(ί*) + σ(4U/*) for the signature σ(ί] of a 2-periodic null homologous oriented link ί in a closed 3-manifold M in the case of a 2-periodic oriented link in S3. We also show that n(t) = n(ί+) + n(t+ U /*) - 1, where n(ί) denotes the nullity of an oriented link i and /* denotes the knot /* with an arbitrary orientation.