Cyclic Branched Covers and the Goeritz Matrix

Abstract

Most of this chapter will be concerned with a study of the twofold cyclic cover X ₂ → S ³ branched over an n component link L. The link L does not need to be oriented for this to make sense, but it will be sometimes convenient to select an arbitrary orientation in order to consider a Seifert surface. The principle result here is that the order of the first homology group H ₁ (X ₂) is det L—the determinant of the link, where det L = |Δ _L (−1)|—and that this number is often easy to calculate. As will be explained, the link determinant is, up to sign, the determinant of any Goeritz matrix [34] of the link, a matrix which is easy to write down starting from any diagram of the link.