Def/Domain topograph

Logical Connections

This definition logically relies on the following definitions and statements: Def/Lax superbasis, Def/Lax basis, Def/Lax vector, State/Every lax vector in a lax basis is primitive, State/Every lax basis is contained in exactly two lax superbases, Def/Finite sequence, State/Every primitive lax vector belongs to a lax basis, State/Solutions to homogeneous linear Diophantine equations can be found with the LCM

The following statements and definitions logically rely on the material of this page: Def/Range topograph, and State/The domain topograph has no circuits

To visualize the logical connections between this definition and other items of mathematical knowledge, you can visit any of the following clusters, and click the "Visualize" tab: Clust/Binary quadratic forms

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 $f_3$.

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 $p_i$.

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.