Algebraic and topological aspects of the singular twin group and its representations

TL;DR

This work constructs the singular twin group $ST_n$ as a natural analogue of the singular braid group and develops both algebraic and topological viewpoints. It achieves a complete classification of complex homogeneous $2$-local representations of $ST_n$ for all $n\ge 2$, presenting explicit families and detailed irreducibility criteria. The results provide a foundational representation-theoretic framework for a novel algebraic object intertwined with doodle topology, and they open avenues for studying linearity, faithfulness, non-local representations, and potential invariants for (singular) doodles. Overall, the paper advances understanding of how singularity-augmented braid-like structures can be represented and analyzed.

Abstract

In this article, we introduce the singular twin monoid and its corresponding group, constructed from both algebraic and topological perspectives. We then classify all complex homogeneous $2$-local representations of this constructed group. Moreover, we study the irreducibility of these representations and provide clear conditions under which irreducibility holds. Our results give a structured approach to understanding this new algebraic object and its representations.

Figures (6)

• Figure 1: The generators $\sigma_i$ and $\sigma_i^{-1}$.
• Figure 2: The generator $s_i$.
• Figure 3: The generators $\tau_i$ and $\tau_i^{-1}$.
• Figure 4: The local moves $D_{1}$ (left) and $D_{2}$ (right).
• Figure 5: A singular doodle obtained as an embedding of a 4-valent graph. The underlying graph is a disjoint union of 2 two-bouquet graphs.

Theorems & Definitions (22)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Definition 5
• Definition 6
• Definition 7
• Definition 8
• Definition 9
• Definition 10