The Signature and Determinant of a Link in RP3

Abstract

The Gordon-Litherland pairing GF of a surface F generalizes the symmetrized Seifert pairing by allowing F to be nonorientable. The pairing GF is developed for surfaces in real projective 3-space RP3, leading to signature and determinant invariants of links L ⊆ RP3. The set of spanning surfaces of L (i.e. surfaces in RP3 bounding L) is partitioned into two classes by an equivalence relation called S∗-equivalence. It is shown that only one of these classes contains orientable surfaces. Consequently, two distinct signature and determinant invariants arise. This contrasts the case of links in S3, where the pairing GF determines a unique signature and determinant, and the case of links in thickened surfaces, where signatures and determinants come in unordered pairs. Explicit computational methods are given.