Unprovability results involving braids

2007/11/23 by Lorenzo Carlucci, Patrick Dehornoy, Andreas Weiermann · 1 citation
Mathematics · #Advanced Combinatorial Mathematics #Algebra over a field #Algebraic number #Algebraic structures and combinatorial models #Braid #Braid group #Braid theory #Computer science #Construct (python library) #Geometric and Algebraic Topology #Mathematical analysis #Mathematics #Order (exchange) #Programming language #Pure mathematics #Simple (philosophy) #math.LO #msc:03B30 #msc:03F35 #msc:20F36 #msc:91A50

paper · pdf · doi:10.1112/plms/pdq016

32 pages

arxiv created 2007/11/23 · openalex publication_date 2010/06/23 · arxiv updated 2014/02/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We construct long sequences of braids that are descending with respect to the standard order of braids (‘Dehornoy order’), and we deduce that, contrary to all usual algebraic properties of braids, certain simple combinatorial statements involving the braid order are not provable in the subsystems I Σ 1 or I Σ 2 of the standard Peano system (although they are provable in stronger systems of arithmetic).

Citations

• Orderings of mapping class groups after Thurston
• Accessible Independence Results for Peano Arithmetic
• Theory of Braids
• Subsystems of Second Order Arithmetic

Cited by

• Braid group [wikipedia]

Related

• Ordering the braid groups
• Searching for non-order-preserving braids algorithmically
• On Braid Words and Irreflexivity
• The well-ordering of dual braid monoids
• Solutions of the braid equation and orders