The Gordon-Litherland pairing and its many applications

TL;DR

The paper presents a clear, diagrammatic proof of the Gordon-Litherland theorem: the knot signature satisfies $\\sigma(K)=\\operatorname{sign}(G)-\\mu(K)$, where $G$ is the Goeritz matrix and $\\mu(K)$ is a diagram-dependent correction term. Central to the argument is the Gordon–Litherland pairing $\\mathcal{G}_F$ on $H_1(F)$ for any spanning surface $F$, realized via the twofold orientation cover and the transfer map, with the Euler number linking $\\mathcal{G}_F$ to $e(F)$. By leveraging Kirby calculus and Yasuhara’s $S^*$-equivalence, the authors show the combination $\\sigma(M_{\\widehat{F}})+\\tfrac{1}{2}e(F)$ is invariant across spanning surfaces, yielding the main equality. The survey of applications highlights powerful bounds and geometric interpretations across Gordian distances, crosscap numbers, nonorientable slice genus, and the interface with categorification and manifold generalizations, underscoring the unifying role of the Gordon–Litherland pairing in low-dimensional topology.

Abstract

Gordon and Litherland's paper $\textit{On the Signature of a link}$ introduced a bilinear form that simultaneously unifies both the quadratic forms of Trotter and Goeritz. This remarkable pairing of combinatorics and topology has had widespread application in low-dimensional topology. In this expository note, we give a picture proof (via Kirby diagrams) of their main result and discuss the numerous ways their theorem has been put to good use.

Figures (13)

• Figure 1: The knot $7_6$, a Seifert surface for $7_6$, and the same Seifert surface isotoped into disc-band form showing the homology generators $\alpha_1,\alpha_2,\alpha_3,\alpha_4$.
• Figure 2: A checkerboard coloring of $7_6$ and its black spanning surface isotoped into disc-band form.
• Figure 3: Definitions of incidence number and type of a crossing.
• Figure 4: Obtaining a basis for the black surface (left) and calculating the Gordon-Litherland form relative to that basis.
• Figure 5: The four possible pairs $(\eta(C),\text{type}(C))$ for a crossing $C$.

Theorems & Definitions (12)

• Theorem 1.1: Gordon-Litherland
• Example 2.1
• Definition 3.1: Gordon-Litherland form
• Example 3.2
• proof : Proof of step (3)
• Theorem 3.3: Yasuhara yasuhara
• Remark 3.4
• Theorem 4.1: Murakami-Yasuhara murakami_yasuhara
• Theorem 4.2: Yasuhara yasuhara_96
• Theorem 4.3: Gilmer-Livingston gilmer_livingston