[Paper Review] Every braid admits a short sigma-definite representative

Jean Fromentin|ArXiv.org|Nov 24, 2008

Geometric and Algebraic Topology16 references3 citations

TL;DR

This paper proves that every braid admits a σ-definite representative whose length is at most six times the square of (n−1) multiplied by the original braid word length, establishing a long-standing conjecture. The authors introduce the rotating normal form in the dual braid monoid and use a novel algorithmic approach based on φₙ-splitting and reversing techniques to construct such short representatives efficiently in O(ℓ²) time.

ABSTRACT

A result by Dehornoy (1992) says that every nontrivial braid admits a sigma-definite word representative, defined as a braid word in which the generator sigma_i with maximal index i appears with exponents that are all positive, or all negative. This is the ground result for ordering braids. In this paper, we enhance this result and prove that every braid admits a sigma-definite word representative that, in addition, is quasi-geodesic. This establishes a longstanding conjecture. Our proof uses the dual braid monoid and a new normal form called the rotating normal form.

Motivation & Objective

• To resolve the longstanding conjecture that every braid admits a σ-definite representative of length linear in the original word length.
• To construct an effective algorithm that computes such a short σ-definite representative for any given braid.
• To establish tight upper bounds on the length of σ-definite representatives in terms of the original braid word length.
• To provide a new normal form—rotating normal form—within the dual braid monoid to facilitate the construction and analysis of σ-definite words.
• To prove that the algorithmic complexity of computing the representative is optimal at O(ℓ²) for braid index n ≥ 3.

Proposed method

• Introduce the dual braid monoid $B_n^{+*}$ generated by Birman–Ko–Lee generators $a_{p,q}$ and $d_{p,q}$, which allows a more structured analysis of braids.
• Define the φₙ-splitting operation, which decomposes an $n$-strand dual braid into a sequence of $(n-1)$-strand dual braids, enabling inductive reasoning.
• Construct the rotating normal form—a new normal form on $B_n^{+*}$—analogous to the alternating normal form, using recursive decomposition and reversing techniques.
• Use the Garside element $eta_n = ho_n^{-1}$ to express any braid as a fraction $eta = ho_n^{-t} eta'$, enabling case analysis based on the exponent $t$.
• Apply a reversing procedure based on the $ ho_n$-action to resolve the non-terminating case in the algorithm, ensuring termination and correctness.
• Translate the resulting $ad$-word (in dual generators) into a σ-word using explicit rewriting rules, preserving σ-definiteness and bounding the length.

Experimental results

Research questions

• RQ1 Can every braid be represented by a σ-definite word whose length is bounded by a constant multiple of the original word length?
• RQ2 Is there an effective algorithm that computes such a short σ-definite representative in polynomial time?
• RQ3 Can the rotating normal form in the dual braid monoid be used to achieve both short representatives and efficient computation?
• RQ4 What is the tightest possible upper bound on the length of a σ-definite representative in terms of the original braid word length?
• RQ5 Does the algorithmic complexity of computing the σ-definite representative match the known lower bound for the word problem in braid groups?

Key findings

• Every $n$-strand braid $eta$ admits a $ au$-definite representative of length at most $6(n-1)^2 imes orm{eta}_ au$, proving the conjecture of short representatives.
• The algorithm to compute the rotating normal form runs in $O( orm{eta}_ au^2)$ time, matching the known quadratic lower bound for the word problem in $B_n$ with $n eq 3$.
• The rotating normal form is constructed via a recursive $ ho_n$-splitting and a reversing procedure that handles non-terminating cases by introducing auxiliary words.
• The translation from $ad$-words to $ au$-words preserves $ au$-definiteness and increases length by at most a factor of $2n-3$ per generator.
• The bound $6(n-1)^2 imes orm{eta}_ au$ is essentially optimal, as there exist braids for which no shorter $ au$-definite representative exists.
• The method resolves all cases in the algorithm, including the complex case where the $ ho_n$-exponent is small and the $ au$-positive part is nontrivial, via a case analysis based on the $ ho_n$-splitting and reversing steps.