Def/Domain topograph



=Geometry at a point=

Suppose that $p$ is a point in the topograph, i.e., $p = \{ f_1, f_2, f_3 \}$ is a lax superbasis. It follows that every unordered pair of distinct elements of $p$ is a lax basis. It follows directly that every point $p$ as above is incident to exactly three edges: $$e_1 = \{ f_2, f_3 \}, e_2 = \{ f_3, f_1 \}, \mbox{ and } e_3 = \{f_1, f_2 \}.$$

Since every lax vector in a lax basis is primitive, it follows that every point $p$ is incident to exactly three faces: $f_1$, $f_2$, and $f_3$.

=Edge geometry=

Suppose that $e$ is an edge in the topograph, i.e., $e = \{ f_1, f_2 \}$ is a lax basis. Since every lax vector in a lax basis is primitive, it follows that every such edge $e$ is incident to exactly two faces: $f_1$ and $f_2$.

In addition, every lax basis is contained in exactly two lax superbases, namely, $e$ is contained in the following two superbases: $$p_+ = \{ f_1, f_2, f_1 + f_2 \}, \mbox{ and } p_- = \{ f_1, f_2, f_1 - f_2 \}.$$ Here, the lax vectors $f_1 \pm f_2$ are not individually well-defined (they each depend on choices of representative vectors), however, the unordered pair $\{ p_+, p_- \}$ is well-defined.

It follows that every edge $e$ is incident to exactly two points. We call these the endpoints of $e$.

Paths
Suppose that $p,q \in P$ are points in the topograph. A path from $p$ to $q$ is a finite sequence $e_1, \ldots, e_\ell$ of edges in the topograph, and a finite sequence $p_0, \ldots, p_{\ell}$ of points in the topograph, such that:
 * $p = p_0$.
 * $q = p_\ell$.
 * For all natural numbers $i$, satisfying $1 \leq i \leq \ell$, the edge $e_i$ has endpoints $p_{i-1}$ and $p_i$.

=Face geometry= Suppose that $f$ is a face in the topograph, i.e., $f = [a,b]$ is a primitive lax vector. Since every primitive lax vector belongs to a lax basis, it follows that there exist $c,d \in \ZZ$, such that $g = [c,d]$ is a primitive lax vector and $e = \{f, g \} $ is an edge incident to $f$.

From the definition of lax bases, it follows that the edges incident to $f$ are in natural bijection with the solutions $g' = (c',d')$ to the linear Diophantine equation: $$ad' - bc' = 1.$$ In particular, $ad - bc = 1$, since $e = \{f,g \}$ is a lax basis.

Given such a solution $g' = (c',d')$ to this linear Diophantine equation, we find that: $$a(d - d') - b(c - c') = 0.$$ Since this is a homogeneous linear Diophantine equation, and $GCD(a,b) = 1$ (so that $LCM(a,b) = ab$), we find that there exists an integer $n \in \ZZ$ such that: $$c = c' + an, \mbox{ and } d = d' + bn.$$ Conversely, every integer $n \in \ZZ$ yields a solution to the Diophantine equation via the above formula. It follows that:
 * The set of edges incident to the face $f$ is in bijection with the set of integers. Namely, there exists a sequence of edges $e_n$, indexed by $n \in \ZZ$, and a sequence $g_n$, indexed by $n \in \ZZ$, such that:
 * $e_n = \{ f, [g_n] \}$ is a lax basis, i.e., an edge incident to $f$.
 * $g_n = g + n f$.
 * Every edge incident to $f$ is equal to $e_n$ for some $n \in \ZZ$

Observe that each edge $e_n$ is "connected" to the edges $e_{n-1}$ and $e_{n+1}$, via the points $\{ f, [g_n], [g_{n+1}] \}$ and $\{ f, [g_n], [g_{n-1}] \}$, respectively.