Restricted Reidemeister Moves in Knot Diagrams

Updated 29 November 2025

Restricted Reidemeister moves are specific subsets of classical knot moves that impose constraints on orientation, type, or sign to isolate key diagrammatic features.
They enable the derivation of minimal generating sets and complexity bounds by selectively restricting moves to achieve refined combinatorial classifications.
Algorithmic applications of these moves yield monotonic simplification procedures and novel invariants that quantify knot complexity and uncover structural obstructions.

A restricted Reidemeister move is any subset of the classical local moves on knot or link diagrams—Ω₁ (type I), Ω₂ (type II), and Ω₃ (type III)—subject to a constraint, such as orientation, direction, sign, combinatorial restriction, or planar/topological context. The study of such restricted move-sets has driven advances in minimal generating sets, complexity bounds, classification problems, and the computation of diagram invariants across both classical and generalized knot theories. The purpose of restriction is typically to isolate the structural roles played by subsets of moves, to combinatorially characterize simplified or algorithmic equivalence classes, or to reveal new diagrammatic or algebraic obstructions that full isotopy ignores.

1. Foundational Definitions: Forms of Restricted Reidemeister Moves

Restricted Reidemeister moves appear in several canonical forms, each precisely defined by the classes of allowed operations:

Flat and Oriented Restrictions:
- Flat moves consider only the local topology of diagrams as generic immersions $S^1 \to S^2$ (monogons, bigons, triangles) and ignore over/under structure. The flat moves $R_1$ , $R_2$ , $R_3$ act as the analogues of standard $\Omega_1$ , $\Omega_2$ , $\Omega_3$ , but without orientation or crossing data (Ito et al., 2020).
- Oriented and Directed Moves: Each move is split by orientation and traversal direction (increasing/decreasing crossings), leading to 16 oriented or 32 directed moves, as catalogued for all knot and tangle diagrams (Suwara, 2016, Ito et al., 25 Nov 2025).
Move-Type and Sign Restrictions:
- Limits may be imposed to only allow moves of certain types, or type-and-sign (e.g., only positive $\Omega_1$ , or only matched/unmatched $\Omega_2$ ).
- Coherent vs. Braid-Type: Coherent (e.g., $2c$, $2d$) vs. braid-type ($2a$, $2b$) moves, and non-braid-type (e.g., $3a$, $3h$) vs. braid-type ($3b$–$3g$) third moves, are essential in classifying minimal generating sets (Ito et al., 25 Nov 2025).
Contextual and Topological Constraints:
- Restrictions can be global or local, such as forbidding moves near a marked point (as in centered diagrams and the crossing number additivity theorem (Weinstein, 22 Nov 2025)), on substructures in rectangular diagrams (Ando et al., 2014), or on projections to non-planar surfaces or spines (Petronio, 17 Jun 2025).
Generalized/Hybrid Moves:
- Certain results introduce “generalized” moves (e.g., $\Omega_3^*$ which allows arbitrary rectangles along the legs of a type-III move (Sasaki, 2022)), or new move-forms such as “composite detection” in algorithmic settings (Petronio et al., 2015).

2. Classification Theorems and Minimal Generating Sets

Restricted move-sets are a core subject of classification theorems that establish when subsets suffice to generate the full diagrammatic equivalence (knot or link isotopy):

Polyak’s Theorem and Variants: Polyak established that $\{\Omega_{1a},\Omega_{1b},\Omega_{2a},\Omega_{3a}\}$ form a minimal generating set of oriented moves. Subsequently, directed versions yield that eight directed Polyak moves suffice for all oriented, directed moves (Suwara, 2016).
Minimal Generating Set Classifications:
- For the full 16 oriented moves, all minimal generating sets of size 4 or 5 have been classified except for four 4-element candidates; any such set must contain precisely one type-III, two type-II, two type-I moves (or fall into a special coherent configuration) (Ito et al., 25 Nov 2025).
- For rotational diagrams with extra up/down constraints, the minimal generating set counts are 8 (unframed) and 9 (framed), with the full descriptions of allowed moves tabulated (Becerra et al., 18 Jun 2025).
- In the presence of topological constraints (e.g., composite diagrams), the set of allowed “restricted” moves may be enlarged with sliding or forbidden configurations, but always remain within explicit combinatorial bounds (Weinstein, 22 Nov 2025).
Role of Ascending/Descending and Braiding Structure: Certain knots (notably, the figure-eight) force the need for both ascending and descending $\Omega_3$ moves; without both, not all isotopies can be realized (Suwara, 2016).

3. Algorithmic Implications and Monotonic Simplification

Restricting moves to subsets aligning with algorithmic or computational objectives leads to monotonic simplification procedures and new algorithmic invariants:

Monotonic Algorithms: By using only non-increasing crossing number moves (e.g., restricted $Z_1$ , $Z_2$ , $Z_3$ generalizations of the classical $R_1$ , $R_2$ , $R_3$ ), one obtains procedures that always reduce or maintain the crossing number. These methods empirically perform efficiently on both trivial and nontrivial knots, detect composite structure, and avoid known combinatorial bottlenecks of the unrestricted calculus (Petronio et al., 2015).
Complexity and Additivity: Weinstein proved that for composite knots, when only restricted (non-sliding) moves are used around a “critical point” of the diagram, the crossing number of $K_0\#K_1$ is at least $c(K_0)+c(K_1)$ . This relies on forbidden moves crossing specified diagrammatic domains and matrix-combinatorial arguments (Weinstein, 22 Nov 2025).
Rectangular and Cromwell Moves: Algorithmic frameworks for arc-presentations and rectangular diagrams require explicit tracking of interior/exterior merges and exchanges, leading to $O(n^2)$ –type explicit upper bounds for their realization in Reidemeister moves (Ando et al., 2014).

4. Diagrammatic Invariants and Obstruction Theory

Restricted move-sets produce new “diagrammatic” invariants—quantities constant under the allowed moves, but which can change under forbidden ones. These invariants provide sharp lower bounds, obstructions to sorting moves, and quantify complexity in knot homotopy:

Flat and Weak Homotopy Invariants: Under flat $R_1+R_2$ moves, projections are classified by their unique reduced (monogon-free, bigon-free) representatives (Ito et al., 2020). For weak $(1,3)$ homotopy (R₁ + only one “positive” R₃), the positive-resolution homomorphism $p(P)$ into oriented knot isotopy classes allows for detection of when a projection can be trivialized without using the unavailable signatures of the missing move-direction (Ito et al., 2020).
Additive Type-II Invariants: Quandle state-sum invariants constructed from $n$ -up–down colorings and cocycles are preserved by all but type-II moves, providing lower bounds for necessary type-II moves and yielding detection in both classical and virtual knot categories (Oshiro et al., 2017).
Arnold-Type and Gauss-Diagram Invariants: For unmatched RII moves, $J^-/2+\mathrm{St}$ and its writhe-corrected variants yield exact counts for sequences that must necessarily use unmatched RIIs (as in the Hayashi et al. example with $r=2+\cos(n\theta/(n+1))$ (Hayashi et al., 2010)). Gauss-diagram formulas for 2-component links detect the necessity of type-II moves where standard perestroika theory fails (Ito, 2022).

5. Structural and Topological Contexts for Restricted Moves

Generalizing the setting of diagrams to singular surfaces, spines, and other 3-manifold presentations demands extensions and further restrictions:

3-Manifold Spines and Flow-Spines: Diagrams projected to almost-special spines or branched flow-spines in arbitrary $3$-manifolds require augmented local move-sets, including edge/vertex-crossing, finger, and half-twist moves. The extended Reidemeister theorem in these contexts is generated by the planar $R_1,R_2,R_3$ together with a small, explicit set of peripheral moves specific to the spine structure (Petronio, 17 Jun 2025).
Twisted Knot Theory and Forbidden Moves: In twisted (generalized virtual) knot theory, the addition of forbidden Gauss diagram slides (F1–F4) is necessary and sufficient to reduce any diagram to trivial form. Each forbidden move corresponds to combinatorial obstructions which cannot be eliminated by generalized Reidemeister moves alone (Xue et al., 2021).

6. Obstructions, Non-Sortability, and Future Directions

Certain configurations demonstrate that not all equivalences can be realized with ordered or non-interleaved restricted sequences:

Sorting and Non-Sortability: There exist pairs of trivial-knot diagrams related by type-I and type-III moves for which no sequence of the form (all I-increase, then all III, then all I-decrease) exists, demonstrating irreducible obstruction phenomena entirely due to the restricted move-set (Sasaki, 2022). The introduction of generalized type-III moves ( $\Omega_3^*$ ) restores sortability, but not in the stricter regime.
Classification and Generalization Open Problems: The minimal generating set problem is fully resolved for all but four explicit sets of the 16 four-move candidates, with further progress depending on the detection of new diagrammatic or link invariants (Ito et al., 25 Nov 2025). The additivity of crossing number for composite knots under unrestricted moves remains an open problem, but is validated for several restricted move-sets (Weinstein, 22 Nov 2025).

7. Summary Table: Key Classes of Restricted Reidemeister Moves

Move Restriction	Allowed Moves	Key Invariant/Property
Flat (projection)	$R_1$ , $R_2$ , $R_3$ (no over/under)	Reduced immersion classification, flat homotopy classes
Oriented/Directed	$\Omega_{1,2,3}^*$ (with direction)	Minimal generating sets, forward/backward invariants
Weak (1,3) homotopy	$R_1$ + positive $R_3$ only	Positive-resolution map $p(P)$ , kernel characterizes triviality
Monotonic simplification	$Z_1,Z_2,Z_3$ , $C$ (restricted)	Non-increasing crossing number; algorithmic minimality
Type/I/III only	$\Omega_1,\Omega_3$ (possibly restricted)	Sortability obstructions; generalized move necessity
Band/spine/3-manifold	$R_{1,2,3}$ + spine-specific moves	Complete local calculus for diagrams in singular topologies

The systematic study of restricted Reidemeister moves unifies combinatorial, algorithmic, and geometric approaches in knot theory, revealing the deep dependence of diagrammatic, algebraic, and complexity-theoretic phenomena on the precise local move-sets permitted. Major applications include sharp lower/upper bounds on move counts, explicit generative descriptions, and the rigorous understanding of equivalence in spaces far beyond the planar case. Continued research aims to close the classification of minimal generating sets, deepen the role of move-specific invariants, and extend topological diagram calculus to ever-more complex ambient settings.