[Paper Review] On the singular braid monoid

V. V. Vershinin|arXiv (Cornell University)|Sep 20, 2003

Geometric and Algebraic Topology9 references4 citations

TL;DR

This paper establishes Garside's foundational results—particularly the existence of a greedy normal form and a solution to the word problem—for the singular braid monoid, extending Birman-Ko-Lee's presentation of the braid group to this singular setting. The key contribution is proving that the singular braid monoid admits a Garside structure via a fundamental word $Δ$ that commutes with singular generators $x_i$ in a controlled way, enabling algorithmic solutions to the word problem and normal forms.

ABSTRACT

Garside's results and the existense of the greedy normal form for braids are shown to be true for the singular braid monoid. An analogue of the presentation of J. S. Birman, K. H. Ko and S. J. Lee for the braid group is also obtained for this monoid.

Motivation & Objective

• To extend Garside’s theory of the braid group—specifically the existence of a greedy normal form and solvability of the word problem—to the singular braid monoid $SB_n$.
• To establish an analogue of the Birman-Ko-Lee presentation for the singular braid monoid using generators $a_{ts}$ and $b_{ts}$, generalizing the braid group’s structure.
• To demonstrate that the fundamental word $Δ$ in the singular braid monoid behaves analogously to its role in the braid group, particularly in commuting with singular generators $x_i$ via $Δ x_i \doteq x_{n-i} \u0394$.
• To provide a constructive, Garside-style solution to the word problem in $SB_n$ using positive equivalence and divisibility relations, avoiding the technical complexity of prior approaches.

Proposed method

• Adapts Garside’s original arguments for the braid group to the singular braid monoid, focusing on the positive singular braid monoid $SB_n^+$ with generators $\sigma_i, x_i$ and relations excluding inverses.
• Introduces a fundamental word $\delta$ defined as $\delta \equiv a_{n(n-1)}a_{(n-1)(n-2)}\cdots a_{21}$, which is shown to be positively equivalent to words beginning or ending with any generator $a_{ts}$, enabling divisibility arguments.
• Uses the Malcev rule and induction on word length to prove cancellation and reduction rules for positive words, particularly for cases involving $\sigma_i A \doteq \sigma_k B$, $\sigma_i A \doteq x_k B$, and $x_i A \doteq x_k B$, establishing unique normal forms.
• Proves commutation relations between the fundamental word $\delta$ and generators $a_{ts}$, $b_{ts}$, showing $a_{ts}\delta \doteq \delta a_{(t+1)(s+1)}$ and $b_{ts}\delta \doteq \delta b_{(t+1)(s+1)}$ for $1\leq s<t<n$, generalizing Birman-Ko-Lee’s results.
• Demonstrates that the relations of the Birman-Ko-Lee presentation for the braid group are consequences of the singular braid monoid’s relations, validating the extension of their framework.
• Applies geometric intuition and symbolic manipulation to verify that the singular braid monoid inherits the Garside property, including the existence of a greedy normal form via left- and right-divisibility.

Experimental results

Research questions

• RQ1 Does the singular braid monoid $SB_n$ admit a Garside structure, including a fundamental word $\Delta$ and a greedy normal form?
• RQ2 Can the word problem in $SB_n$ be solved using a Garside-style algorithm, analogous to the braid group?
• RQ3 Is there an analogue of the Birman-Ko-Lee presentation for the singular braid monoid, using generators $a_{ts}$ and $b_{ts}$?
• RQ4 How does the fundamental word $\delta$ in the singular braid monoid interact with singular generators $x_i$, and does it satisfy $x_i \delta \doteq \delta x_{n-i}$?
• RQ5 Are the relations of the Birman-Ko-Lee presentation derivable from the singular braid monoid’s defining relations?

Key findings

• The singular braid monoid $SB_n$ admits a greedy normal form, extending Garside’s theory to this singular setting.
• The fundamental word $\delta$ is positively equivalent to any word beginning or ending with a given generator $a_{ts}$, enabling divisibility-based normal form construction.
• The fundamental word $\delta$ commutes with generators $a_{ts}$ and $b_{ts}$ via $a_{ts}\delta \doteq \delta a_{(t+1)(s+1)}$ and $b_{ts}\delta \doteq \delta b_{(t+1)(s+1)}$ for $1\leq s<t<n$, generalizing the braid group’s commutation relations.
• The word problem in $SB_n$ is solvable via a Garside-style algorithm, with cancellation and reduction rules for positive words involving $\sigma_i$, $x_i$, and their combinations.
• The Birman-Ko-Lee presentation for the braid group is extended to the singular braid monoid, with all their relations derivable from the singular braid monoid’s defining relations.
• The singular braid monoid inherits the Garside property: every element has a unique normal form, and the fundamental word $\delta$ serves as a Garside element, ensuring solvability of the word problem and existence of a normal form.