Manuscript En03-A@1 · En03-A.tex · proposition · 5398614e

Proposition: Exact ordered products

External conical geometry and certified local moves › Four affine faces · lines 1685–1712 · proof included or linked

En03-A@1#prop:R6, En03-A@1#eq:incoming-products, En03-A@1#eq:outgoing-products

Every closed contractible core word is the affine identity. In particular,

Λ_{E \leftarrow W} Λ_{W \leftarrow N} Λ_{N \leftarrow E} Λ_{N \leftarrow S} Λ_{S \leftarrow E} Λ_{E \leftarrow N} = I, = I, Λ_{E \leftarrow W} Λ_{W \leftarrow S} Λ_{S \leftarrow E} Λ_{N \leftarrow S} Λ_{S \leftarrow W} Λ_{W \leftarrow N} = I, = I .

The event four-face word is also $I$ .

Proof. Every word telescopes:

(C_{i_{0}} C_{i_{1}}^{- 1}) \dots (C_{i_{m - 1}} C_{i_{0}}^{- 1}) = I .

Displayed formulas in this passage

En03-A@1#eq:incoming-products · line 1689
$Λ_{E \leftarrow W} Λ_{W \leftarrow N} Λ_{N \leftarrow E} Λ_{N \leftarrow S} Λ_{S \leftarrow E} Λ_{E \leftarrow N} = I, = I, Λ_{E \leftarrow W} Λ_{W \leftarrow S} Λ_{S \leftarrow E} Λ_{N \leftarrow S} Λ_{S \leftarrow W} Λ_{W \leftarrow N} = I, = I .$
En03-A@1#eq:outgoing-products · line 1689
$Λ_{E \leftarrow W} Λ_{W \leftarrow N} Λ_{N \leftarrow E} Λ_{N \leftarrow S} Λ_{S \leftarrow E} Λ_{E \leftarrow N} = I, = I, Λ_{E \leftarrow W} Λ_{W \leftarrow S} Λ_{S \leftarrow E} Λ_{N \leftarrow S} Λ_{S \leftarrow W} Λ_{W \leftarrow N} = I, = I .$
unlabeled · line 1708
$(C_{i_{0}} C_{i_{1}}^{- 1}) \dots (C_{i_{m - 1}} C_{i_{0}}^{- 1}) = I .$