The Coxeter Quotient of the Braid Group as a Universal Inversion-Symmetrizing Quotient

The Coxeter quotient π: Bₙ → Sₙ of the Artin braid group forgets everything about a braid except the permutation of its strands. We show that this quotient is characterized by a symmetry property: it is the universal quotient of Bₙ on which the inversion involution ι: σᵢ ↦ σᵢ⁻¹ acts trivially, and the full twist Δ²—a 2π rotation—is inverted by ι and therefore frozen out. Among homogeneous additive invariants of the braid monoid, only the mod-2 parity homomorphism survives passage to the quotient. On the other hand, order-theoretic structure survives on the Garside intervals: on the interval of positive permutation braids the braid length coincides with the rank of the weak Bruhat lattice of the frozen records, while on the dual Birman–Ko–Lee interval the rank is reflection length. Together with the classical interval theorems of Elrifai–Morton and Bessis, this gives two canonical lattice orders on the frozen records and two rank functions, which disagree in general. Machine verifications for n = 3 and n = 4 are included.