Factorizing the Brauer monoid in polynomial time

TL;DR

The paper addresses the problem of factoring tangles in the Brauer monoid $\mathcal{B}_{N}$ by introducing a polynomial-time approach. It builds a length function via a polynomial-time mapping $\tau: \mathcal{B}_{N} \to S_N$ that preserves factorization length, and develops two length notions, $\ell_{\tau}$ and $\ell_{P}$, to guide a minimal-factorization procedure. The authors present an $O(N^4)$ factorization algorithm that maintains edge-crossing information efficiently and prove structural properties (node polarity, pass-through/come-back constraints, LZR decomposition) to justify the method. They report empirical validation of key assumptions, provide a concrete implementation, and discuss potential optimizations and open combinatorial questions, highlighting the method's significance for scalable Brauer-tangle analysis and related RNA-structure mappings.

Abstract

Finding a minimal factorization for a generic semigroup can be done by using the Froidure-Pin Algorithm, which is not feasible for semigroups of large sizes. On the other hand, if we restrict our attention to just a particular semigroup, we could leverage its structure to obtain a much faster algorithm. In particular, $\mathcal{O}(N^2)$ algorithms are known for factorizing the Symmetric group $S_N$ and the Temperley-Lieb monoid $\mathcal{T}\mathcal{L}_N$, but none for their superset the Brauer monoid $\mathcal{B}_{N}$. In this paper we hence propose a $\mathcal{O}(N^4)$ factorization algorithm for $\mathcal{B}_{N}$. At each iteration, the algorithm rewrites the input $X \in \mathcal{B}_{N}$ as $X = X' \circ p_i$ such that $\ell(X') = \ell(X) - 1$, where $p_i$ is a factor for $X$ and $\ell$ is a length function that returns the minimal number of factors needed to generate $X$.

Figures (18)

• Figure 1: On the left a tangle $X = (1,3)(2,1')(2',3')$ in $\mathcal{B}_{3}$, on the right its unique minimal factorization $T_1 \circ U_2$.
• Figure 2: The four cases for which we define the merge operation. In all cases, $e = (x,y)$ and $h = (i,i+1)$. (a) $e$ is an upper hook. (b) $e$ is a lower hook. (c) $e$ is a negative transversal. (d) $e$ is a positive transversal.
• Figure 3: Diagram for the proof of \ref{['thm:tauMinimal']}. Assume $F$ is minimal and $\tau(F)$ is not. Axioms 11 and 13 applied to $\tau(F)$ have preimage to axioms 11-16 applied to $F$. In the end, $\tau(F)$ will contain a $T_i \circ T_i$ because we assumed it was not minimal, which implies $F$ was not minimal too and thus we have a contradiction.
• Figure 4: From a tangle in $\mathcal{B}_{N}$ we obtain a minimal factorization $F$, we then compute $\tau(F)$ which corresponds to a tangle in $S_N$.
• Figure 5: A tangle along with one of its minimal factorizations $F = T_1 \circ U_2 \circ U_3 \circ T_1 \circ T_2$. The edge $(1,3)$ passes through $T_1$ and $U_2$, edge $(2,3')$ passes through two $T_1$s and one $T_2$, edge $(4,1')$ passes through $U_3, U_2$ and $T_1$ and edge $(2',4')$ passes through $T_2$ and $U_3$. Edge $(2,3')$ comes back at the bottom-most $T_1$ because it decreases in size, this also means that it comes back in $F$. Note also that $(2,3')$ is the only edge that passes through only $T$-primes and satisfies the property $\#T(e) \geq |e|$.

Theorems & Definitions (38)

• Definition 1
• Theorem 1
• proof
• Corollary 1
• proof
• Corollary 2
• Corollary 3
• proof
• Corollary 4
• proof