Manuscript VII-E@1 · VII-E.tex · proposition · b713cf3c

Proposition: Local ordered products

One certified four-face slicing exchange › Adapted charts and exact products · lines 2070–2104 · proof included or linked

VII-E@1#prop:products

Every closed core transition word is the identity. In particular, the incoming words

Λ_{E \leftarrow W} Λ_{W \leftarrow N} Λ_{N \leftarrow E}, Λ_{E \leftarrow W} Λ_{W \leftarrow S} Λ_{S \leftarrow E}

and the outgoing words

Λ_{N \leftarrow S} Λ_{S \leftarrow E} Λ_{E \leftarrow N}, Λ_{N \leftarrow S} Λ_{S \leftarrow W} Λ_{W \leftarrow N}

equal $I$ . The four-face event word also equals $I$ .

Proof. Every word is a product

(C_{i_{0}} C_{i_{1}}^{- 1}) (C_{i_{1}} C_{i_{2}}^{- 1}) \dots (C_{i_{m - 1}} C_{i_{0}}^{- 1}),

which telescopes.

Displayed formulas in this passage

unlabeled · line 2074
$Λ_{E \leftarrow W} Λ_{W \leftarrow N} Λ_{N \leftarrow E}, Λ_{E \leftarrow W} Λ_{W \leftarrow S} Λ_{S \leftarrow E}$
unlabeled · line 2084
$Λ_{N \leftarrow S} Λ_{S \leftarrow E} Λ_{E \leftarrow N}, Λ_{N \leftarrow S} Λ_{S \leftarrow W} Λ_{W \leftarrow N}$
unlabeled · line 2098
$(C_{i_{0}} C_{i_{1}}^{- 1}) (C_{i_{1}} C_{i_{2}}^{- 1}) \dots (C_{i_{m - 1}} C_{i_{0}}^{- 1}),$