Toroidal Knotoids: Diagrammatic and Invariant Approaches

• Toroidal knotoids are open-ended knot diagrams on a torus, defined as immersed intervals with designated endpoints and characterized by both local Reidemeister moves and global toroidal isotopies.
• They extend classical knot theory by incorporating lifting procedures into the thickened torus, establishing a clear correspondence between diagrammatic moves and rail isotopy.
• Generalized bracket polynomials and skein module structures provide robust invariants for classifying toroidal knotoids, with applications ranging from polymer physics to field theory.

A toroidal knotoid is a generalization of a classical knotoid diagram, which itself is a knot diagram with open endpoints, placed on the torus (the oriented surface $T^2 = S^1 \times S^1$) rather than in the plane, disc, or sphere. The theory of toroidal knotoids unifies topological, diagrammatic, and algebraic perspectives and plays a key role in understanding open-ended and knotted structures in settings where the ambient space has nontrivial topology. The recent literature has focused on equivalence, moves, lifting procedures, invariant polynomials, and connections with the skein module theory of 3-manifolds, as well as links to field theoretic and physical realizations.

1. Definition and Equivalence for Toroidal Knotoids

A toroidal knotoid is defined as an oriented immersed interval in the torus $T^2$ with finitely many transverse double points (crossings), each equipped with over/under data. The endpoints retain their role as the "leg" and "head," analogously to planar or annular knotoids. The equivalence relation is generated by the usual three Reidemeister moves (R1, R2, R3) applied away from the endpoints, but in the toroidal setting, the ambient isotopy group includes additional "global" moves that arise from the nontrivial topology:

• Longitudinal Moves: Sliding arcs along a meridian, allowing an endpoint to move across a noncontractible cycle.
• Meridional Moves: Sliding arcs along the longitude, similarly transporting the endpoint across the complementary cycle.

These toroidal moves, in addition to local Reidemeister ones, generate the full equivalence of toroidal knotoid diagrams (Diamantis et al., 5 Sep 2025). The extra flexibility inherent in the torus is essential: two diagrams equivalent in $T^2$ may not be equivalent in an annulus or disc because the ambient isotopy class detects the toroidal geometry.

2. Lifting and Rail Isotopy in the Thickened Torus

A central methodological tool in the theory is lifting a toroidal knotoid into a spatial setting. Each toroidal knotoid diagram $K$ can be embedded into the thickened torus $T^2 \times I$. Crossings are localized in disjoint 3-balls, and arcs between crossings are replaced by embedded arcs away from the "rails," which are two parallel segments perpendicular to the surface on which the endpoints reside. Two such lifts are rail isotopic if they are related by an ambient isotopy (rel rails) that allows the endpoints to slide along the prescribed rails (Diamantis et al., 5 Sep 2025). The key result is that rail isotopy in $T^2 \times I$ corresponds exactly to the diagrammatic equivalence relation on $T^2$.

Furthermore, lifting procedures create a bridge to the paper of skein-theoretical and quantum invariants, since skein modules are naturally defined in thickened surfaces. This lifting is analogous to the passage from diagrams to spatial knots but incorporates the open endpoints and toroidal ambient data.

3. Extension of Bracket Polynomials and Invariants

The bracket polynomial for toroidal knotoids generalizes the Kauffman–Turaev loop bracket from the planar/annular case to $T^2$:

$$\langle K \rangle = \sum_{s \in S(K)} A^{\sigma_s} d^{k_s} v^{m_s} s_{p_s,q_s}^{l_s}$$

where $S(K)$ is the set of all states of $K$ (i.e., all complete smoothings of crossings), $A$ is the usual skein variable, $d = -A^2 - A^{-2}$, $v$ is a variable tracking loops that encircle the trivial segment (if present), $s_{p,q}$ is an indeterminate associated to each essential (noncontractible) curve of type $(p,q)$, $l$ is the number of such components, and $\sigma_s, k_s, m_s$ record the smoothing and component data (Diamantis et al., 5 Sep 2025).

A universal version promotes $s_{p,q}^l$ to a family $s_{p,q,l}$ and similarly for $v^m$ to $v_m$. A reduced version specializes $s_{p,q}$ to monomials $x^p y^q$. The normalization by $(-A^3)^{-\operatorname{w}(K)}$ yields a toroidal Jones-type invariant.

This construction recovers in the toroidal context the Kauffman bracket skein module structure for $T^2 \times I$, since the states are parameterized by collections of essential curves (with specific winding data) and null-homotopic curves, together with the residual arc from the knotoid. The bracket polynomial thus distinguishes not only local crossing information but captures the essential winding characteristics unique to $T^2$ (Diamantis et al., 5 Sep 2025).

4. Mixed Knotoids, Braidoids, and Diagrammatic Structures

The concept of mixed knotoids is employed to systematically encode toroidal (and annular) knotoids as planar diagrams with additional fixed components:

• Mixed Knotoids: Diagrams in $S^2$ comprising a moving knotoid part and a fixed unknotted component (representing the complementary solid torus for annular knotoids or a Hopf link for toroidal knotoids). This representation enables leveraging classical techniques, such as skein relations and bracket polynomials, to the toroidal setting (Diamantis, 2021, Diamantis et al., 5 Sep 2025).
• Braidoids and L-moves: Mixed braidoids are "toroidalized" analogues of classical braids, with the moving part augmented by a fixed identity strand. The Markov and Alexander theorems extend: any mixed knotoid is isotopic to the closure of a mixed braidoid, and their closure equivalence is governed by L-moves and isotopy (Diamantis, 2021).

This approach enables the formulation of toroidal knotoid invariants and skein modules, framed diagrammatically via mixed diagrams and closure operations.

5. Inclusion Relations: Planar, Annular, and Toroidal Knotoids

The natural embeddings of surfaces yield a hierarchy in the theory:

Surface	Embedding Relation	Diagrammatic Effect
Disc	$\subset$ Annulus	Planar $\to$ Annular
Annulus	$\subset$ Torus ($T^2$)	Annular $\to$ Toroidal

These inclusions induce injection maps between respective knotoid theories, with bracket polynomials and state-sum formulas respecting the embedding (Diamantis et al., 5 Sep 2025). For instance, a planar knotoid may acquire additional equivalence in the torus due to global toroidal moves, while annular knotoids embed as toroidal ones that do not exploit both longitudinal and meridional winding. The containment is reflected both at the diagrammatic and invariant level.

6. Connections to Field Theory and Physical Realizations

Toroidal knotoids have parallels in field theory. Knotted domain strings on toroidal host solitons in $3+1$ dimensions can be constructed using scalar fields with a $\mathbb{Z}_2$ Wess–Zumino-type potential (Eto et al., 2012). Under certain choices of potential, single (meta-)stable domain strings may occur with free endpoints and general boundary behavior, directly realizing topological structures akin to toroidal knotoids. The (p,q)-winding classification for torus knots reflects the fundamental pair of winding numbers for knotoids embedded in $T^2$.

A plausible implication is that future adapted field-theoretic models, such as those employing a sine-Gordon-type potential, could realize stable toroidal knotoid defects with free endpoints. The mapping from theoretical soliton constructions to the algebraic and topological invariants of toroidal knotoid diagrams offers avenues for both mathematical classification and physical application.

7. Applications and Future Directions

The classification of toroidal knotoids via bracket polynomials and skein module theory provides a new toolkit for the paper of open knotted structures in handlebodies, thickened surfaces, and field systems with nontrivial ambient topology. Applications are anticipated in:

• The tangling and topology of polymer chains and molecular scaffolds on toroidal substrates (Barthel et al., 2015).
• The classification of domain wall intersections and open string defects in condensed matter and high-energy physics, where open-ended knotted structures naturally arise (Eto et al., 2012).
• The extension to pseudo knotoids and other generalizations incorporating ambiguous crossing data, leveraging the weighted resolution set invariants (Diamantis et al., 5 Sep 2024).

A plausible implication is that the toroidal knotoid framework can be further extended to periodic knotoids, tangloids, or multi-component open-ended links in higher-genus surfaces, and that richer skein modules will emerge as central classification tools.

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By synthesizing the combinatorial, topological, and algebraic aspects, the theory of toroidal knotoids integrates surface topology, diagrammatic moves, state-sum invariants, and physical realizability, providing a robust platform for the exploration of open-ended topological phenomena in mathematics and physics.