Annular Multi-Knotoids

• Multi-knotoids of the annulus are topological objects generalizing classical knots by using immersed intervals and circles with specialized moves in a nontrivial surface.
• The framework employs a four-variable annular bracket polynomial and extended skein theory to derive invariants that distinguish complex isotopy classes.
• Applications span protein topology and tangle theory, linking annular, planar, and toroidal perspectives while enabling advanced combinatorial and algebraic classifications.

Multi-knotoids of the annulus are topological objects generalizing classical knots and knotoids by allowing diagrams in an annular surface with multiple open and closed components. Building on the foundational knotoid theory developed by Turaev, the paper of multi-knotoids in the annulus has catalyzed a convergence of combinatorial, algebraic, and geometric techniques. This framework accommodates the annulus' nontrivial topology and supports a suite of extended invariants and skein-theoretic tools, enabling a finer classification of topological types and their behavior under isotopy, closure, and interaction with the ambient 3-manifold structure.

1. Definitions and Diagrammatic Equivalence

A multi-knotoid of the annulus is an equivalence class of diagrams in the annulus $\mathcal{A}$, each consisting of a finite collection of immersed intervals (open components with two endpoints called "leg" and "head") and possibly several disjoint immersed circles (closed components). The fundamental equivalence relation is generated by:

• Surface isotopies of the annulus,
• “Swing moves” (allowing endpoints to move within their boundary regions but not over/under other arcs),
• The classical Reidemeister moves R1, R2, R3 away from endpoints.

This extends knotoid diagram equivalence from the planar and spherical settings into the annulus context, introducing essential and nontrivial cycles unobservable in simply connected environments. The endpoint behavior and the annulus’ noncontractible nature lead to new isotopy classes and permitted moves, formalized in constructions such as the lifting of diagrams to the thickened annulus $\mathcal{A} \times I$, where endpoints are rooted on parallel “rails” perpendicular to the core of the annulus (Diamantis et al., 5 Sep 2025).

2. Relationships to Planar and Toroidal Knotoids

There exist natural inclusion maps among planar, annular, and toroidal knotoid theories:

• The inclusion of a disk into the annulus provides an injection from planar knotoids to annular knotoids.
• The annulus itself embeds in the torus, and thus annular knotoids serve as a subclass within toroidal knotoids. This hierarchy, with the annulus as an intermediate setting, is crucial for lifting constructions and for understanding how isotopy and closure operations behave under surface changes (Diamantis et al., 5 Sep 2025).

A systematic technique involves “mixed knotoids”: planar knotoid diagrams augmented by a fixed unknot (the $O$–component) for annular knotoids, or a fixed Hopf link for toroidal knotoids. The mixed moves constitute an algebraic and combinatorial dictionary between planar diagrams and their annular or toroidal interpretations, crucial for translating combinatorial invariants and isotopy relations into the more complex ambient topology.

3. Bracket Polynomials and Skein-Theoretic Invariants

The extension of state-sum invariants to the annulus requires distinguishing between types of closed state components:

• Null-homotopic (contractible) loops,
• Essential loops (those not contractible, traversing the core of the annulus)—divided into “inner” and “outer” based on their relation to the open component,
• Nesting loops (non-essential loops that still enclose the trivial knotoid).

For an annular multi-knotoid, the bracket polynomial generalizes the Turaev loop bracket by associating variables $x$, $v$, and $t$ to inner essential, nesting, and outer essential loops, respectively, after smoothing all crossings. This produces a four-variable Laurent polynomial in $A$, $x$, $v$, $t$. The explicit evaluation rules, as given in (Diamantis et al., 5 Sep 2025), assign multiplicities to smoothed diagrams:

$\langle\,\text{state}\,\rangle = x^n\, v^m\, t^l$

where $n$, $m$, $l$ count the respective loop types, and the normalization assigns $1$ to the trivial arc. The universal version further refines this by introducing infinitely many variables to index the loop contributions:

$$\langle\cdot\rangle_U \in \mathbb{Z}\left[A^{\pm1},\,\{x_n\},\,\{v_m\},\,\{t_l\}\right]$$

This universal annular bracket polynomial realizes the Kauffman bracket knotoid skein module of the thickened annulus, encoding the complete algebraic structure arising from the state-sum evaluation (Diamantis et al., 5 Sep 2025).

4. Lifting and Annular Mixed Knotoids

Lifting an annular knotoid involves embedding its diagram into the thickened annulus $\mathcal{A} \times I$, with endpoints attached to perpendicular rails. This 3-dimensional perspective provides a direct topological model for interpreting state-sum invariants, isotopy relations, and closure operations. The concept of $O$-mixed knotoids reconstructs the annular context using planar diagrams coupled with a fixed component representing the co-core of the annulus. Mixed Reidemeister moves supplement classical moves by governing interactions between moving and fixed parts, creating a unified framework for algebraic and topological paper.

Additionally, pseudo knotoids and tied pseudo knotoids have been generalized to the annulus, where pseudo crossings accommodate undefined over/under information, and “ties” encode extra combinatorial relations. For pseudo knotoids, an Alexander-type theorem asserts that every pseudo (multi-)knotoid is isotopic to the closure of a pseudo braidoid, with equivalence classified by $L$-moves and braid isotopies (Diamantis, 2020).

5. Invariants for Distinction and Classification

Multi-knotoids of the annulus admit a suite of refined invariants:

• Homological Casson-type invariant: By transforming a knotoid diagram from $S^2$ to the annulus (removing disks around endpoints), the associated first homology group $H_1(\mathcal{A}) \simeq \mathbb{Z}$ provides a setting for the invariant $CH^\wedge$, which encodes subgroup data of loops arising from crossings. This invariant refines the integer-valued Casson-type numbers and provides sharp lower bounds on crossing number and criteria for non-classicality (“proper” knotoids) (Tarkaev, 2020).
• Kauffman Bracket Skein Module: Extended to multi-linkoids (unions of open intervals and closed circles in the annulus), these skein modules track both contractible and essential cycles, with the module freely generated by crossingless diagrams without trivial components (Gabrovšek et al., 2022).
• Quandle cocycle invariants: The shadow quandle cocycle invariant, calculated by summing local weights determined by 2- or 3-cocycles over colored diagrams, distinguishes between multi-knotoids and their mirrors, detects chirality, and yields lower bounds on minimal crossing number (Cazet, 2022).
• Annular pseudo bracket and Jones polynomials: For pseudo links and their annular counterparts, the bracket and Jones polynomials are defined as four-variable Laurent polynomials, with the extra variable $s$ marking essential (noncontractible) loops (Diamantis et al., 1 Jan 2025). In the mixed link approach, multi-knotoids in the annulus correspond to $O$-mixed pseudo links in $S^3$, and associated invariants are computed accordingly.

6. Connection to Graph Polynomials and Extended Theorems

The extension of the Thistlethwaite theorem to multi-knotoids of the annulus proceeds via the construction of marked ribbon graphs from state diagrams. The bracket or arrow polynomial of a knotoid (or a multi-knotoid/linkoid) is then evaluated by a corresponding Bollobás–Riordan polynomial assigned to the ribbon graph. For knotoids in the annulus or more general compact surfaces, twisted arrow polynomials and loop arrow polynomials further enrich the invariant structure and align with topological features of the ambient surface (Chmutov et al., 16 Dec 2024). These developments facilitate algorithmic computations and connect knotoid invariants to a broader graph-theoretic context.

7. Applications, Classifications, and Future Directions

The framework of multi-knotoids in the annulus has significant implications:

• In protein topology and modeling open-chain molecules, annular knotoid invariants quantify entanglement with consistency under ambiently relevant topological operations (Chmutov et al., 16 Dec 2024).
• For tangle theory, results about closures of 1-tangles and annulus twists (utilizing the wrapping index and sutured manifold theory) illuminate the structure and uniqueness of possible knotoid closures in an annular environment; for instance, at most two distinct closures of a nontrivial 1-tangle can yield the unknot, tightly tied to the annular ambient topology (Taylor, 12 Jun 2025).
• Universal skein-theoretic formulations position annular multi-knotoids as a bridge between classical knot theory and higher-genus or virtual knot settings.
• Potential extensions include categorification, analysis of annular instanton Floer homology and spectral sequences approximating annular Khovanov homology for multi-knotoids, and explorations of quantum invariants in this richer context (Xie, 2018).

Future research is expected to expand the interplay between algebraic, combinatorial, and geometric approaches to classify multi-knotoid types, understand their relationships to spatial graph structures (e.g., generalized theta graphs), and deepen the application of topological invariants enabled by the annulus’ inherent cyclic structure.