Manuscript VIII-ST@1 · VIII-ST.tex · section · fe979f3b

Cells and exact finite metric data

An explicit persistent transverse cellulation › Cells and exact finite metric data · lines 533–569

VIII-ST@1#eq:born, VIII-ST@1#eq:faces

Let $P_{s}$ be the transverse apex intersection. Persistent vertices have spatial coordinates

A_{i} = r e_{i}, D_{i} = 4 e_{i}, τ (A_{i}) = τ (D_{i}) = s + 2.

For $s > 0$ , additional vertices are

B_{i} = β s e_{i}, C_{i} = χs (e_{i} + e_{i + 1}), τ (B_{i}) = 2 s, τ (C_{i}) = 2 s + b χs .

Spatial symbols stand for their lifted graph points when used in faces.

For $s \leq 0$ , the core face in sector $i$ is

V_{i} = [P, A_{i}, A_{i + 1}], τ = s + a (ξ + η) .

For $s > 0$ , use

U_{i}^{-} U_{i}^{+} V_{i} = [P, B_{i}, C_{i}], = [P, C_{i}, B_{i + 1}], = [B_{i}, A_{i}, A_{i + 1}, B_{i + 1}, C_{i}], τ τ τ = 2 s + b η, = 2 s + b ξ, = s + a (ξ + η) .

The $V_{i}$ may be nonconvex. For all $s$ , retain

Q_{i} = [A_{i}, D_{i}, D_{i + 1}, A_{i + 1}], τ = s + 2,

and the resting exterior beyond $ρ = 4$ .

Displayed formulas in this passage

VIII-ST@1#eq:born · line 542
$B_{i} = β s e_{i}, C_{i} = χs (e_{i} + e_{i + 1}), τ (B_{i}) = 2 s, τ (C_{i}) = 2 s + b χs .$
VIII-ST@1#eq:faces · line 557
$U_{i}^{-} U_{i}^{+} V_{i} = [P, B_{i}, C_{i}], = [P, C_{i}, B_{i + 1}], = [B_{i}, A_{i}, A_{i + 1}, B_{i + 1}, C_{i}], τ τ τ = 2 s + b η, = 2 s + b ξ, = s + a (ξ + η) .$
unlabeled · line 537
$A_{i} = r e_{i}, D_{i} = 4 e_{i}, τ (A_{i}) = τ (D_{i}) = s + 2.$
unlabeled · line 553
$V_{i} = [P, A_{i}, A_{i + 1}], τ = s + a (ξ + η) .$
unlabeled · line 565
$Q_{i} = [A_{i}, D_{i}, D_{i + 1}, A_{i + 1}], τ = s + 2,$