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

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.

💡 Research Summary

The paper introduces a novel algebraic and topological object called the singular twin group, denoted (ST_n), together with its monoid counterpart (STM_n). Building on the well‑known braid group (B_n) and its singular extension (SB_n), as well as the twin group (T_n) (the “flattened” version of braids where over/under information is forgotten), the authors define (STM_n) by adjoining singular generators (\tau_i) to the twin generators (s_i). The defining relations combine the twin relations (s_i^2=1) and commutation for distant indices with the singular braid relations, yielding equations (10)–(14). By adjoining inverses of the (\tau_i) they obtain the singular twin group (ST_n), a natural extension of (T_n) analogous to the passage from (B_n) to (SB_n).

A substantial part of the work is devoted to the pure singular twin subgroup (SPT_n), defined as the kernel of the natural epimorphism (ST_n\to S_n). Using the Reidemeister–Schreier method, the authors compute an explicit presentation for the case (n=3), showing that (SPT_3) is generated by four elements (a=s_1\tau_1), (b=s_2\tau_2), (c=s_2 a s_2), and (d=s_1 b s_1). This concrete computation illustrates that (ST_n) is a regular extension of the symmetric group.

The core representation‑theoretic contribution is the complete classification of complex homogeneous 2‑local representations of (ST_n). A (k)-local representation (Definition 7) maps each generator to a block matrix with an identical (k\times k) block (M). For (k=2) the authors translate all defining relations of (ST_n) into matrix equations, obtaining a system that forces (M) to satisfy a specific polynomial condition. Consequently, every homogeneous 2‑local representation is parametrized by a complex parameter (t) (or equivalently by the eigenvalues of (M)), and the authors list all admissible families.

Section 5 investigates the irreducibility of these representations. By examining invariant subspaces and eigenvalue multiplicities, they prove that the representation is irreducible precisely when the parameter avoids a finite exceptional set (e.g., (t\neq 0,\pm2)). When (t) takes one of these special values, the representation decomposes, and a distinguished ((n-1))-dimensional invariant factor appears, mirroring results known for the twin group’s N₁ and N₂ representations. The paper thus extends the irreducibility criteria to the singular setting.

Topologically, the authors interpret elements of (ST_n) as singular doodles: immersions of 4‑valent graphs (possibly together with disjoint circles) into the 2‑sphere, allowing transverse double points (from the (\tau_i)) and 4‑valent vertices (from the (s_i)). They describe local moves D₁–D₄ that generate the equivalence relation, analogous to Reidemeister moves for knots. This provides a geometric picture linking the algebraic presentation to a class of planar objects.

The paper concludes with several open directions: extending the classification to (k>2) local representations, computing (co)homology and K‑theoretic invariants of (ST_n), and developing a full theory of singular doodles and their move equivalence. Overall, the work establishes the singular twin group as a natural bridge between braid‑type algebraic structures and low‑dimensional topology, delivering a thorough algebraic presentation, a concrete pure subgroup description, a full classification of homogeneous 2‑local representations, and clear irreducibility criteria.