Mixed Knotoids: Open Curves with Fixed Anchors

• Mixed knotoids are knotoid diagrams that include an open curve along with fixed closed components, capturing the topology of ambient 3-manifolds.
• They enable planar representations of annular and toroidal knotoids via fixed anchors, extending classical invariants like the bracket polynomial.
• They transform complex topological problems into combinatorial ones by applying extended Reidemeister moves and skein module techniques.

A mixed knotoid is a knotoid diagram in a surface—most commonly the plane or the 2-sphere—that incorporates both an "open" knotoid component (an immersed oriented interval with endpoints) and one or more fixed closed components (typically an unknot or a Hopf link), whose presence encodes the topology of the ambient 3-manifold. The concept has become central to the extension of knotoid theory beyond simply connected planar or spherical ambient spaces, allowing planar representations to model annular and toroidal knotoids via the addition of fixed “anchor” components. Mixed knotoids thus enable the transfer of results and invariants from the planar framework to knotoids in the solid torus, the thickened annulus, or the thickened torus, as well as the efficient paper of topological entanglement in open curves under complex ambient topologies (Diamantis et al., 5 Sep 2025, Diamantis, 2021).

1. Foundational Principles and Definition

Mixed knotoids are a generalization of knotoids, which themselves are immersions of an interval $[0,1]$ (with endpoints denoted leg and head) into an oriented surface, with finitely many transverse double points equipped with navigation data at crossings. Ordinary knotoids are studied up to ambient isotopy and local Reidemeister moves performed away from endpoints, with forbidden moves tightly restricting endpoint behavior (Turaev, 2010).

In the mixed setting, additional fixed closed components are included:

• O-mixed knotoids: Planar (or spatial) multi-knotoid diagrams with a pointwise fixed unknot $O$. The fixed unknot models the boundary of the complementary solid torus when passing from a planar setting to the thickened annulus.
• H-mixed knotoids: Multi-knotoid diagrams that include a fixed Hopf link $H$ (two fixed circles linked once), encoding the topology of the thickened torus when moving from a planar context to the toroidal case (Diamantis et al., 5 Sep 2025).

The essence of a mixed knotoid is the coexistence of a moving (knotoid) part and a fixed "topological anchor," such as $O$ or $H$, which remains invariant under all ambient isotopies of the "moving" part but is essential to the ambient topology.

2. Topological Interpretation and Inclusion Relations

The motivation for mixed knotoids emerges from the modeling of knotoids in more complex 3-manifolds—such as the solid torus (annular knotoids) or thickened tori (toroidal knotoids)—by embedding them as planar entities that retain a record of the ambient topology via a fixed component.

• In annular knotoids, the planar diagram is mixed with a fixed unknot $O$. The O-mixed diagram remains in $S^3$, with $O$ marking the complementary solid torus; this allows annular knotoids in the thickened annulus to be modeled as O-mixed knotoids (Diamantis et al., 5 Sep 2025).
• For toroidal knotoids, a fixed Hopf link $H$ is used, as the complement in $S^3$ is homeomorphic to a thickened torus. The inclusion relations are formalized via injections and surjections corresponding to embeddings of the disk in the annulus and of the annulus in the torus. However, surjectivity may be lost due to torus-specific moves such as the longitudinal toroidal move, which have no annular analogue (Diamantis et al., 5 Sep 2025, Diamantis, 2021).

Lifting techniques enable annular and toroidal knotoids to be studied through their planar mixed representatives, reducing topological problems in complex settings to combinatorial problems in the plane but with additional rules encoding the complexity of the ambient manifold.

3. Mixed Knotoid Moves and Isotopy

Mixed knotoid diagrams are considered up to an extended set of equivalence moves:

• Classical Reidemeister moves performed solely on the moving (open) part, away from endpoints;
• Mixed Reidemeister moves (generalized R2 and R3) that involve strands of the moving part interacting with the fixed component (see Figures "mreidm" and "almo" in (Diamantis, 2021));
• Endpoint moves allowing the leg or head to pass over or under the fixed part;
• Fake forbidden moves recognizable only in the presence of the fixed component;
• Braid-theoretic analogues such as mixed braidoids and corresponding $L$-moves, enabling algebraic manipulation of diagrams equivalent to the topological setting (see (Diamantis, 2021) and (Diamantis et al., 5 Sep 2025)).

Combined, these moves provide a diagrammatic equivalence that respects both the local entanglement of the open curve and the ambient topology encoded by the fixed component(s).

4. Bracket Polynomials and Skein Module Realization

Extending Turaev's framework for the bracket polynomial of knotoids (Turaev, 2010), mixed knotoids demand bracket polynomials capable of detecting the manifold's topology:

• Planar Knotoid Bracket:

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

with $d = -A^2 - A^{-2}$, and $S(K)$ denoting the set of states, each with parameters counting the number of loops of certain types.

• Annular Bracket:

$$\langle K \rangle = \sum_{s\in S(K)} A^{\sigma_s} d^{k_s} x^{n_s} v^{m_s} t^{l_s}$$

with exponents counting inner, nesting, and outer loops according to their position relative to the fixed $O$ (Diamantis et al., 5 Sep 2025).

• Toroidal Bracket:

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

where $(p_s, q_s)$ is the slope type for each essential toroidal loop and $s_{p,q}$ represents a polynomial variable for such a component (Diamantis et al., 5 Sep 2025).

These universal analogues recover the Kauffman bracket skein module of the thickened annulus or torus, and thus encode the full data of isotopy classes under the relevant equivalence (Diamantis et al., 5 Sep 2025).

5. Theoretical and Practical Implications

The mixed knotoid formalism has several key implications:

• Reduction Principle: Problems about knotoids in thickened annuli or tori are reduced to planar (mixed) knotoid diagrams, with additional moves and variables prescribing the effect of ambient topology.
• Transfer of Invariants: The bracket polynomial formalism (and thus also the Jones polynomial and related quantum invariants) can be extended coherently, allowing, for example, Turaev’s loop bracket to be realized for annular and toroidal cases—something not immediate in a purely diagrammatic planar theory (Diamantis et al., 5 Sep 2025).
• Modeling of Physical Systems: Mixed knotoids used as proxies for annular and toroidal knotoids allow analytical and computational paper of entanglement in open protein chains, open polymers, and periodic structures where open and closed strands coexist and interact (Goundaroulis et al., 2017, Dorier et al., 2018, Gabrovšek et al., 2022).
• Generalization to Multi-component Settings: The framework extends naturally to multi-knotoids and linkoids, where both open and closed components coexist, often necessary for biological or polymeric applications.

6. Extensions and Open Research Directions

Potential extensions and open problems include:

• Classification Problems: The precise relation between annular, toroidal, and planar (mixed) knotoids remains to be fully classified, particularly concerning the injectivity or surjectivity of the inclusions between their respective isotopy classes (Diamantis et al., 5 Sep 2025).
• Skein Theory & Quantum Invariants: The explicit realization of bracket skein modules via mixed knotoid diagrams presents opportunities for new quantum invariants for open curves in thickened surfaces, including potential connections to categorified invariants and quantum algebra.
• Generalized Surfaces and Nonorientable Contexts: Techniques from mixed knotoids can be adapted to knotoids on non-orientable surfaces or in the presence of further algebraic constraints (Chmutov et al., 16 Dec 2024).
• Periodic and Lattice Models: The mixed framework offers a foundation for the rigorous mathematical treatment of doubly periodic tangloids and related open structures in materials science (Diamantis et al., 5 Sep 2025).

7. Representative Formulas and Diagrammatic Conventions

To summarize key formulas:

Type	Bracket Polynomial Formula	Variables
Planar	$$\langle K \rangle = \sum_{s} A^{\sigma_s} d^{k_s} v^{m_s}$$	$d = -A^2 - A^{-2}$, $v$ counts nesting loops
Annular	$$\langle K \rangle = \sum_{s} A^{\sigma_s} d^{k_s} x^{n_s} v^{m_s} t^{l_s}$$	$x$ = inner, $v$ = nesting, $t$ = outer loops
Toroidal	$$\langle K \rangle = \sum_{s} A^{\sigma_s} d^{k_s} v^{m_s} s_{p_s,q_s}^{l_s}$$	$s_{p,q}$ labels toroidal loop of slope $(p,q)$

These state sum expansions—indexed by the appropriate loop and arc types—provide the foundation for the calculation of invariants within the mixed knotoid framework. Diagrammatically, mixed knotoids are drawn as planar diagrams with a fixed closed component (or components) preserved under all isotopies, and the open knotoid(s) regarded as subject to standard and mixed Reidemeister moves.

---

Mixed knotoids thus function as a bridge between the combinatorics of planar diagrams and the topology of more general 3-manifolds, providing a robust, computable setting for the development of quantum invariants, the paper of open curve entanglement, and the systematic exploration of skein module representations for a wide spectrum of topological contexts (Diamantis et al., 5 Sep 2025, Diamantis, 2021).