The Gordon–Litherland pairing for links in thickened surfaces

Abstract

We extend the Gordon–Litherland pairing to links in thickened surfaces, and use it to define signature, determinant and nullity invariants for links that bound (unoriented) spanning surfaces. The invariants are seen to depend only on the [Formula: see text]-equivalence class of the spanning surface. We prove a duality result relating the invariants from one [Formula: see text]-equivalence class of spanning surfaces to the restricted invariants of the other. Using Kuperberg’s theorem, these invariants give rise to well-defined invariants of checkerboard colorable virtual links. The determinants can be applied to determine the minimal support genus of a checkerboard colorable virtual link. The duality result leads to a simple algorithm for computing the invariants from the Tait graph associated to a checkerboard coloring. We show these invariants simultaneously generalize the combinatorial invariants defined by Im, Lee and Lee, and those defined by Boden, Chrisman and Gaudreau for almost classical links. We examine the behavior of the invariants under orientation reversal, mirror symmetry and crossing change. We give a 4-dimensional interpretation of the Gordon–Litherland pairing by relating it to the intersection form on the relative homology of certain double branched covers. This correspondence is made explicit through the use of virtual linking matrices associated to (virtual) spanning surfaces and their associated (virtual) Kirby diagrams.

The Jones polynomial from a Goeritz matrix

“…First, it shows that every symmetric, integer matrix is the Goeritz matrix of some link in a thickened surface. This result extends [2, Theorem 3.5]. Theorem Every symmetric, integer matrix $$G$$ is a Goeritz matrix of a checkerboard‐colorable link diagram in a thickened surface.…”

A characterization of alternating links in thickened surfaces

“…In this section, we review the Gordon-Litherland pairing [7] and its extension to links in thickened surfaces [1]. The pairing is defined for any link L ⊂ Σ × I that admits a spanning surface, which is a compact, connected surface F embedded in Σ × I with boundary ∂F = L. The surface F may or may not be orientable, and here we consider it as an unoriented surface.…”

“…The Gordon-Litherland pairing is extended to thickened surfaces Σ × I in [1], and we review its definition. Let F ⊂ Σ × I be a compact, unoriented surface without…”

“…Set V = P ∪ γ N . Since P and N are both orientable, V is a closed orientable surface, and 1 2 e(P , γ) = 0 = 1 2 e(N , γ). Therefore, σ P (γ) = sig(G P ) and σ N (γ) = sig(G N ).…”

“…In the case that F is a spanning surface for a link L ⊂ S 3 , the signature of L can be computed in terms of the signature of the pairing together with a correction term. The Gordon-Litherland pairing is extended to Z/2 homology 3-spheres in [8] and to thickened surfaces Σ × I in [1]. Our characterization of alternating links in Σ × I is phrased in terms of the Gordon-Litherland pairing on spanning surfaces for the link.…”