move compatibility section

notes/paper-notes/q4-Orientiation-Equivalence.md

@@ -75,11 +75,6 @@ minimizers of $`ℱ_{\text{LG}}`$ for $`L → 0`$ are "suitably approximated" by
 Q = s(n ⊗ n - \tfrac13 \operatorname{Id}) \quad (s = -3 λ_1 = -3 λ_2, n = ê_3)
 ```
 
-## Compatibility
-- Oseen-Frank and de Gennes are compatible if the oriented line field (= unit vector field) can be oriented (without changing regularity)
-  - otherwise Oseen-Frank might miss a global minimizer because it is not orientable
-  - Question if orientable calculable with integer programming problem (p. 4 = 496) (= (linear) optimization problem with only integer coefficiants)
-
 ### Notation
 - $`P : 𝕊^2 → 𝒬`$ removes orientation. Orientable = in the image of $`P`$. Same for $`Q`$ only defined on $`∂Ω`$
 - $`𝒬 := \{Q = s(n ⊗ n - \frac13 \operatorname{ID}) | n ∈ 𝕊^2\}`$
@@ -172,6 +167,16 @@ Let $`Q ∈ W^{1,p}(Ω, 𝒬), 1 ≤ p ≤ ∞`$ be non-orientable. Then there e
 - specific energy functionals are regarded (p. 11/503) because they were looked at before and Oseen-Frank was successful for them
   - conversion between energy functionals possible
 
+
+## Compatibility
+
+- Oseen-Frank and de Gennes are compatible if the oriented line field (= unit vector field) can be oriented (without changing regularity)
+  - otherwise Oseen-Frank might miss a global minimizer because it is not orientable
+  - Question if orientable calculable with integer programming problem (p. 4 = 496) (= (linear) optimization problem with only integer coefficiants)
+- In chapter 3, orienting a line field is translated into lifting $`Q`$ from $`ℝP^2`$ to covering space $`𝕊^2`$.
+  - Classical algebraic topology tools only consider continous maps, but we have $`W^{1,p}(Ω)`$ maps and want to preserve this regularity
+
+
 ## Questions
 
 ## Errors in the paper