Singular Twin Monoid

• Singular Twin Monoid is an algebraic structure that extends twin groups by introducing singular generators to model marked double points in topological doodles.
• It incorporates precise generator-relator relations that blend swap operations and singular interactions, parallel to structures in braid and pseudo braid monoids.
• Its rigorous local and homogeneous representation theory provides new invariants and classification tools for studying symmetric singularities in low-dimensional topology.

The singular twin monoid arises in the algebraic and topological paper of structures generalizing symmetric and braid groups, specifically as an extension of the twin group that incorporates singular crossings. This object encodes both swap-like ("twin") operations and singular behaviors which, in topological settings, correspond to marked double points in diagrams akin to doodles. The singular twin monoid and its corresponding group provide a platform for representing configurations with intertwined symmetry and singularity, and their representation theory reveals a rich landscape of local and homogeneous linear actions.

1. Algebraic Structure and Defining Relations

The singular twin monoid, denoted STMₙ, is constructed by adjoining singular generators to the classical twin group Tₙ. The twin group Tₙ is generated by swap elements $s_1, s_2, \ldots, s_{n-1}$ satisfying $s_i^2 = 1$ and $s_i s_j = s_j s_i$ for $|i-j| \geq 2$. The singular extension introduces generators $\tau_1, \tau_2, \ldots, \tau_{n-1}$ subject to relations:

• $\tau_i \tau_j = \tau_j \tau_i$ for $|i-j| \geq 2$
• $\tau_i s_j = s_j \tau_i$ for $|i-j| \geq 2$
• $\tau_i s_i = s_i \tau_i$
• $s_i s_{i+1} \tau_i = \tau_{i+1} s_i s_{i+1}$
• $\tau_i s_{i+1} s_i = s_{i+1} s_i \tau_{i+1}$

This structure closely parallels the Baez–Birman singular braid monoid, with the distinguishing feature that singular generators $\tau_i$ interact with swap generators rather than the over/under crossings of the braid group. Often, group extensions are formed by adjoining formal inverses to the singular generators.

2. Topological Interpretation: Singular Doodles and Moves

Topologically, the twin group is associated with doodles—immersions of circles into the 2-sphere where crossing data is "flattened" (over/under information is ignored). The singular twin monoid models singular doodles, where singular generators $\tau_i$ correspond to marked double points, or transverse intersections. Besides the basic doodle moves D₁ and D₂ (elementary reductions akin to Reidemeister moves), the singular theory requires additional local moves (D₃ and D₄) to process configurations involving singular vertices.

This connection implies that STMₙ is equipped to classify and encode the combinatorial structures arising in singular doodle theory, capturing both symmetry and intersection signatures.

3. Relation to Virtual, Pseudo, Tied, and Twisted Braid Monoids

Structural similarities and isomorphisms exist between STMₙ and several other diagrammatic monoids:

• An isomorphism between the singular braid monoid SMₙ and the pseudo braid monoid PMₙ holds via the correspondence $\sigma_i \mapsto \sigma_i, \tau_i \mapsto p_i$ (Bardakov et al., 2015). This result implies that algebraic techniques and invariants for SMₙ transfer to pseudo braids and, by analogy, to STMₙ when swap generators play the role of underlying permutations.
• Reduced presentations for related monoids such as the virtual singular braid monoid VSBₙ, where the algebraic complexity is lowered by expressing all but one crossing generator in terms of base generators and virtual permutations (Caprau et al., 2015), suggest similar reduction methods for STMₙ. In particular, permutations (or swap-like elements) play a central role in encoding strand positions and their interactions with singularities.
• Tied singular braid monoids TSₙ incorporate both set partitions (ties, forming idempotents) and the singular braid monoid, resulting in a semidirect product $TS_n \cong P_n \rtimes S_n$ (Arcis et al., 2020, Aicardi et al., 2018). The "singular twin monoid" can be viewed as an algebraic relative when the twin generators are extended with tie and singular generators, leading to refined algebraic frameworks and new invariants for singular links.
• The monoid of singular twisted virtual braids, which includes generators for classical, virtual, singular, and twisting ("bar") elements, features relations expressible through virtual elements. Reduction schemes allow the expression of most generators in terms of a minimal set, paralleling simplification techniques available for STMₙ (Negi et al., 26 Mar 2024).

4. Local and Homogeneous Representation Theory

A core development is the systematic classification of complex homogeneous $k$-local representations ($k=2,3$) of the singular twin group STₙ (Nasser et al., 5 Oct 2025). In these representations, each generator is mapped to a block-diagonal matrix in which the relevant part is a $k \times k$ matrix, identical across locations (homogeneity). Explicitly:

• 2-local representations: For $n \geq 3$, each $s_i$ and $\tau_i$ acts on $\mathbb{C}^n$ via block matrices $M_i$ and $N_i$. Examples:
  • $$M_i = \begin{bmatrix} 0 & b \ 1/b & 0 \end{bmatrix}$$, $$N_i = \begin{bmatrix} w & x \ x/b^2 & w \end{bmatrix}$$
• 3-local representations: For $n \geq 4$, on $\mathbb{C}^{n+1}$,
  • $s_i \mapsto \operatorname{diag}(I_{i-1}, N, I_{n-i-1}), \ N=\begin{bmatrix}0 & b & 0\1/b&0&0\0&0&1\end{bmatrix}$

Irreducibility criteria are precisely stated: for families where $N_i$ has parameters $(w,x,b)$, irreducibility requires $w+(x/b) \neq 1$.

In general, $n=2$ representations are always reducible, while for $n\geq3$ irreducibility is controlled by the absence of common eigenvectors and parameter non-degeneracy (e.g., avoidance of roots of specified polynomials for some families).

5. Applications and Significance

The singular twin monoid and group enable algebraic and topological modeling of doodle and singular doodle configurations, with the algebraic theory precisely capturing the symmetries and intersection structures present in these topological objects. Their representation theory lays the groundwork for constructing invariants analogous to those arising in classical knot and braid theory (e.g., Alexander-type polynomials via Burau-like representations), with potential to distinguish doodles and singular doodles.

The tools developed—especially the explicit homogeneous local representations and their irreducibility conditions—provide a foundation for exploring faithfulness, defining invariants, and studying the category structure underlying singular doodles and their generalizations. These developments promise further advances in classification problems in low-dimensional topology, as well as potential applications in mathematical physics (e.g., quantum computation, statistical mechanics) where symmetric and singular interactions play a role.

6. Connections to Non-Cancellative Monoids and Higher Homotopy

In broader contexts, phenomena such as non-cancellativity in monoids with twin-like structure capture secondary topological information—specifically, the failure of cancellation is associated with nontrivial second homotopy classes (Saito, 2023). For instance, twins of non-cancellative tuples give rise to elements ("$\Pi$-classes") in $\pi_2$ of the underlying space, which obstruct the $K(\pi,1)$ property and differentiate between singular and nonsingular structures. This suggests that singular twin monoids may encode subtle geometric information about configuration spaces or discriminant loci, with algebraic obstructions reflected in topological invariants.

7. Summary Table of Defining Data

Generator	Relation Type	Significance
$s_i$	$s_i^2=1$, $s_is_j=s_js_i$ ($|i-j|\geq2$)	Swap/twin action (base symmetry)
$\tau_i$	$\tau_i\tau_j=\tau_j\tau_i$ ($|i-j|\geq2$)	Singular crossing (marked double points)
Mixed	$\tau_is_j=s_j\tau_i$ ($|i-j|\geq2$); $\tau_is_i=s_i\tau_i$	Local interaction (singular-twin compatibility)
Exchange	$s_is_{i+1}\tau_i=\tau_{i+1}s_is_{i+1}$; $\tau_is_{i+1}s_i=s_{i+1}s_i\tau_{i+1}$	Propagation of singularities across swaps

Conclusion

The singular twin monoid, and its group extension, synthesize swap and singular crossing symmetries into a single algebraic object, with concrete topological motivations and a fully classified finite-dimensional homogeneous local representation theory. The explicit generator-relator architecture, tight connections to non-cancellative monoids and homotopy invariants, and transferable techniques from braid and pseudo braid theory position the singular twin monoid as a central structure in the paper of symmetric singularities, low-dimensional topology, and their representation theory.