Tarcha 3.15 adjacent braid words define the same homotopy quotient class

Proved

TarchaBraids.thm_3_15_adjacent_word_quotient_eq_v1

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

artin-presentationbraid-groupshomotopytarcha

For adjacent generators, the left and right three-half-twist word loops represent the same path-homotopy class because both equal the same outer-rotation class.

Preamble

import Mathlib
import Definitions.Def_TarchaBraids_adjacent_word_loops_v1
import Theorems.Thm_TarchaBraids_thm_3_15_left_word_eq_outer_quotient_v1
import Theorems.Thm_TarchaBraids_thm_3_15_right_word_eq_outer_quotient_v1

Formal statement

namespace TarchaBraids

open BraidsLinksMCG

theorem thm_3_15_adjacent_word_quotient_eq_v1 :
    ∀ {n : ℕ} (i j : Fin (n - 1)) (hji : (j : ℕ) = (i : ℕ) + 1),
      Path.Homotopic.Quotient.mk (leftBraidWordLoop n i j) =
        Path.Homotopic.Quotient.mk (rightBraidWordLoop n i j) := by sorry

end TarchaBraids

Source

Modular remaining adjacent-relation proof for Tarcha 3.15.

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