Coherence of Seifert circles and the height of a diagram

Assume the Axiom of Choice. Let $C,C^{\mathrm{\prime }}$ be two disjoint oriented circles in $S^2$ (Seifert smoothing and Seifert circles of an oriented link diagram) and let $A$ be the annulus they cobound, which exists and has the two circles as its boundary circles by Two disjoint circles in the two-sphere cobound an annulus. Orient $A$ once, a choice of one of its two orientations (Oriented smooth manifolds and oriented charts); this orientation induces a boundary orientation on each of the two boundary circles $C$ and $C^{\mathrm{\prime }}$, and reversing the orientation of $A$ reverses both induced orientations (Induced boundary orientation).

Coherence. The circles $C$ and $C^{\mathrm{\prime }}$ are coherent when, with respect to one, equivalently any, orientation of $A$, the given orientations of $C$ and $C^{\mathrm{\prime }}$ agree with the two induced boundary orientations in opposite senses: one of the given orientations agrees and the other disagrees. Equivalently, the two given oriented circles represent the same element of $H_1(A;\mathbb{Z})$, which is the formulation used in the survey. The two formulations are the classical description of one-dimensional coherent orientation of an annulus, and the equivalence of reversing the orientation of $A$ is immediate because both induced boundary orientations flip, so the relation "opposite senses of agreement" is independent of the chosen orientation. When the given orientations agree with the induced boundary orientations in the same sense (both agree or both disagree), the circles are incoherent.

Height. For an oriented diagram $D$ with Seifert picture $S$ and Seifert circles $C_1,\dots ,C_m$ (Seifert smoothing and Seifert circles of an oriented link diagram), the height of $D$ is the number of distinct unordered pairs $\{i,j\}$ of indices such that $C_i$ and $C_j$ are incoherent:

$h(D):=\mathrm{\#}\{\{i,j\}:1\le i<j\le m,C_i\text{ and }{C}_j\text{ incoherent}\}\mathrm{.}$

Coherence is defined for every pair of Seifert circles of the picture, not only for pairs joined by a signed arc; the signed arcs only record the crossings of the original diagram. In particular $h(D)=0$ means that all pairs of Seifert circles of $D$ are coherent. The axiom of choice is consumed exactly through the annulus lemma, which supplies the annulus $A$ and its two boundary circles; the rest of the definition, including the invariance under reversing the orientation of $A$, is choice-free. Distinct Seifert circles are disjoint, so the definition applies to every one of the finitely many pairs, including pairs with no signed arc between them. The empty and one-circle pictures have height zero.