Prove2Me

The explicit half-twist has the adjacent deck permutation

Proved
TarchaBraids.halfTwist_deck_permutation_of_quotient_v1

by WillR · Sep 21, 2026 · Mathlib 0df444a (Lean v4.33.1)

braid-groupscovering-spaceshalf-twistsmonodromypermutationstarcha

For any symmetric-group quotient-covering structure on the ordered-to-unordered configuration projection, the explicit elementary half-twist has deck permutation equal to the corresponding adjacent transposition.

Preamble
import Mathlib
import Definitions.Def_BraidsLinksMCG_ConfigSpace
import Definitions.Def_TarchaBraids_HalfTwist
import Definitions.Def_TarchaBraids_endpoint_permutation_action_v1
Formal statement
namespace TarchaBraids

open BraidsLinksMCG

theorem halfTwist_deck_permutation_of_quotient_v1 (n : ℕ) (i : Fin (n - 1))
    (hp : IsQuotientCoveringMap (configProj n) (Equiv.Perm (Fin n))) :
    hp.fundamentalGroupToMulOpposite
      (⟨baseOrdered n, rfl⟩ : (configProj n) ⁻¹' {baseUnordered n})
      (halfTwistBraid n i) =
    MulOpposite.op (Equiv.swap (strandIdx i) (strandIdxSucc i)) := by sorry

end TarchaBraids
Source
Tarcha Teorema 3.11 and the explicit half-twist construction: the elementary braid generator interchanges exactly the two adjacent strand endpoints.

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