The Gordon-Litherland pairing for links in thickened surfaces

TL;DR

This paper extends the Gordon-Litherland pairing to links in thickened surfaces, defining new invariants that generalize existing ones and relate to virtual links, with applications in minimal support genus and computational algorithms.

Contribution

It introduces a generalized Gordon-Litherland pairing for links in thickened surfaces and establishes duality, computability, and applications to virtual link invariants.

Findings

Defines signature, determinant, and nullity invariants for links in thickened surfaces.

Proves a duality relating invariants from different spanning surface classes.

Provides an algorithm for computing invariants from Tait graphs.

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 $S^*$-equivalence class of the spanning surface. We prove a duality result relating the invariants from one $S^*$-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.