arxiv: 2605.01976 · v1 · submitted 2026-05-03 · 🧮 math.GT

Recognition: 2 theorem links

· Lean Theorem

A constructive solution to the equivalence problem for knot projectivizations

Javier Mart\'inez-Aguinaga, Sergio de Mar\'ia

Pith reviewed 2026-05-08 19:17 UTC · model grok-4.3

classification 🧮 math.GT

keywords knotsprojectivizationisotopyequivalencelens spacesRP^3algorithm

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The pith

An algorithm produces explicit isotopies connecting any two projectivizations of the same affine knot in RP^3.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows how to build an explicit isotopy connecting any two projectivizations of a given knot from S^3 inside RP^3. This solves the question of whether all such projectivizations are equivalent by giving a step-by-step way to deform one into the other. The construction generalizes to knots in lens spaces L(p,q). Readers would care because it replaces an abstract existence question with a concrete procedure that can be applied to specific examples.

Core claim

We adapt an idea due to A. Hatcher from embedding spaces to describe an algorithm that produces an explicit isotopy between any two given projectivizations of the same affine knot. We introduce the notion of lensification of a knot in any lens space L(p,q) and describe an algorithm that works in that setting, of which RP^3 is a special case. We apply the algorithm to several pairs of knots from the literature for which the equivalence problem was open, finding explicit isotopies.

What carries the argument

The algorithm obtained by adapting Hatcher's embedding-space techniques to produce isotopies between knot projectivizations.

If this is right

The equivalence of any two projectivizations of the same affine knot can be decided by executing the algorithm.
Explicit isotopies are now available for all pairs of projectivizations of a given affine knot.
The procedure extends directly to lensifications of knots inside arbitrary lens spaces L(p,q).
Specific open equivalence questions from the literature receive concrete isotopy witnesses.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The algorithm could be implemented computationally to verify equivalences for knots with crossing numbers beyond manual reach.
Similar constructive adaptations might resolve equivalence problems for knots in other quotient manifolds.
The method supplies a practical tool for exploring the connected components of the space of knots in RP^3.

Load-bearing premise

Hatcher's embedding-space techniques adapt to knot projectivizations in RP^3 and lens spaces in a way that produces a terminating algorithm giving a valid isotopy exactly when the two projectivizations are equivalent.

What would settle it

Finding two projectivizations of the same affine knot for which the algorithm does not terminate or does not produce a valid isotopy between them.

Figures

Figures reproduced from arXiv: 2605.01976 by Javier Mart\'inez-Aguinaga, Sergio de Mar\'ia.

**Figure 1.** Figure 1: Visualization of the different elements in the 3-ball model of L(p, q) = B3/ ∼. The northern and southern hemispheres are denoted by E+ and E−, respectively. The rotation map gp,q is depicted acting on the point x and the reflection map fz is depicted acting on the point gp,q(x). Thus, the points depicted in blue x ∼ fz (gp,q(x)) are identified with each other. We will denote by F the quotient map F : B3 →… view at source ↗

**Figure 2.** Figure 2: Depiction of the curve f that yields the generator [f] of π1(L(p, q)) = Zp and also of H1(L(p, q)) = Zp. It is defined as a p-th fraction of the counterclockwise oriented equatorial curve. It depicts an actual closed loop since its endpoints are identified and it has order-p since its p-times concatenation yields the whole equatorial curve that is contractible within the 3-ball B3/ ∼. i) L˜ does not pass t… view at source ↗

**Figure 3.** Figure 3: Depiction of a knot in a lens space L(8, 1) = B3/ ∼ together with its associated disk diagram (Definition 2.12). Points coming from the northern hemisphere are labelled with positive natural numbers, in order, following the counterclockwise orientation of the equator ∂B2 0 . Points coming from the southern hemisphere are labelled with negative numbers so that each point labelled with −i is identified with … view at source ↗

**Figure 4.** Figure 4: The Figure describes the process of taking the canonical closure (right) of a long knot (left) by attaching a smooth arc disjoint from the interior of B3 to the non-trivial part of the long knot, whose projection onto the diagram does not yield self-intersections. This is well defined up to homotopy and canonically yields a knot in S 3 out of any long knot in R 3 . without creating new self-intersection po… view at source ↗

**Figure 5.** Figure 5: The figure on the left shows a diagram of a closed embedding γ : S 1 → S 3 together with a choice of a point p on its trace. B is an arbitrarily small ball containing p on its interior such that ∂B intersects the trace of γ in exactly two antipodal points. The part of γ outside B is a long embedding γLong(t) : R → R 3 according to Remark 3.8. The figure on the right shows how to obtain a diagrammatic repre… view at source ↗

**Figure 6.** Figure 6: for the visualization of why it represents the class [f q ] in π1(L(p, q))) or check [12, Lemma 4.3] from where the same fact readily follows view at source ↗

**Figure 7.** Figure 7: Visualization in a disk diagram of the deformation (through an isotopy) of the class-q non-local unknot u(t) : S 1 → B3/ ∼ so that it becomes flat in the region t ∈ (− 1 2 , 1 2 ); i.e. u(t) = (t, 0, 0), for t ∈ (− 1 2 , 1 2 ) without introducing self-intersections in its associated disk diagram. On the other hand, take the long knot γLong and consider its (1/2)-shrinking γ 1/2 Long(t). Replace now the fla… view at source ↗

**Figure 8.** Figure 8: Construction of the lensification γLens(t) of a trefoil knot in a lens space L(p, q) (for the particular choice of (p, q) = (8, 3)). The top-left frame depicts the class-q non-local unknot after a deformation that made it flat in the region t ∈ (− 1 2 , 1 2 ). The bottom-left frame depicts the associated long knot after a 1 2 -scale shrinking. The frame on the right shows the lensification γLens(t) built o… view at source ↗

**Figure 9.** Figure 9: Given a knot γ with a diagram, regard it as a framed knot equipped with the blackboard framing. The corresponding loop of knots γθ(t) (Eq. (4)) has constant evaluation at the origin γθ(0) (black marked point in the six frames) with constant derivative d dt (γθ(t))|t=0 (black arrow). It can thus be regarded as a loop where the knot slides rigidly over itself with fixed basepoint and derivative. We have colo… view at source ↗

**Figure 10.** Figure 10: Case 1 in the algorithm: the first branch according to the orientation of the knot in the disk diagram represents an upper branch in the first crossing. The algorithm specifies to pull the corresponding underpass (in red) all under the knot so that it yields an underpass in the other part of the diagram. The picture depicts this situation in the particular case of L(8, 3). Case 2. It represents a lower br… view at source ↗

**Figure 11.** Figure 11: Case 2 in the algorithm: the first branch according to the orientation of the knot in the disk diagram represents a lower branch in the first crossing. The algorithm indicates to pull the corresponding overpass (in red) all over the knot so that it yields an overpass in the other part of the diagram. Once again, this diagram corresponds to a lensification in L(8, 3). different projectivizations of the sam… view at source ↗

**Figure 12.** Figure 12: The problem of whether the multiple projectivizations of the figure-8 knot K1 (top-left) and K2 (top-right) are equivalent has been raised in [11]. The figure depicts an application of the algorithm, which produces an explicit isotopy between the two and thus provides a constructive solution to the problem. Although all arrows are reversible (the figure describes an isotopy), we suggest the reader to foll… view at source ↗

**Figure 13.** Figure 13: The question of whether the projective knots K1 (top-left) and K2 (bottom-right) are isotopic was raised in [15]. The figure depicts an explicit isotopy obtained via the algorithm presented in Section 3.4.3, thus yielding a constructive solution to the question. 4.3. Third application. In our last application we consider a pair of knots introduced in a preliminary version of the article [11], which first … view at source ↗

**Figure 14.** Figure 14: The equivalence problem for the projective knots K1 (top-left) and K2 (bottom-right) was raised in a prepublished version of the work [11]. The figure exhibits an explicit isotopy between the two obtained via the algorithm presented in Section 3.4.3, therefore yielding a constructive solution to the problem view at source ↗

read the original abstract

The problem of whether different projectivizations of the same affine knot $K\subset\mathbb{S}^3$ are equivalent in $\mathbb{R}\mathbb{P}^3$ can be found in [11] and has also been posed as an open question in [15]. In this note we provide a constructive solution to the problem. In particular, we adapt an idea due to A. Hatcher developed in the realm of embedding spaces and we describe an algorithm that produces an explicit isotopy between any two given projectivizations of the same affine knot. More generally, we introduce the notion of lensification of a knot in any lens space $L(p,q)$ and describe an algorithm that works in that more general setting, of which $\mathbb{R}\mathbb{P}^3\simeq L(2,1)$ is a particular instance. Finally, we apply this algorithm to several pairs of knots from the literature for which the equivalence problem was raised as an open question, finding explicit isotopies.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

They supply a concrete algorithm that builds explicit isotopies between equivalent knot projectivizations in RP^3 and lens spaces by adapting Hatcher's embedding-space stratification.

read the letter

The main takeaway is that this note turns an open existence question into a working procedure: given two projectivizations of the same affine knot, the algorithm outputs an isotopy when they are equivalent. The authors adapt Hatcher's embedding-space ideas to RP^3 and general lens spaces L(p,q), lay out the steps, and verify termination via the finite complexity of the stratification. They also run the method on several pairs from the literature that had been left open, producing the isotopies in each case. That constructive delivery is the real advance and the part that will be most immediately usable for explicit checks in geometric topology. The write-up is direct, the examples are concrete, and there is no circularity or reliance on fitted parameters. The central claim holds up on the evidence given. The soft spots are modest. The termination argument depends on the stratification remaining finite in all cases, which the authors assert but could be spelled out with a clearer complexity bound for very high-genus or high-crossing knots. The examples are helpful but stay within modest complexity, so the practical runtime for large instances is not yet mapped. Overall this is a solid, self-contained note aimed at knot theorists who work with projective or lens-space embeddings and need to resolve specific equivalence questions rather than just know they are decidable. A reader in that niche will get a usable tool. I would send it to peer review; the problem is well-posed, the method is new, and the details are inspectable enough to warrant referee attention.

Referee Report

0 major / 3 minor

Summary. The manuscript claims to resolve the open equivalence problem for projectivizations of affine knots in RP^3 (and more generally lens spaces L(p,q)) by adapting Hatcher's embedding-space stratification techniques. It describes a step-by-step algorithm that, given two equivalent projectivizations, explicitly constructs an isotopy between them, proves termination via finite complexity of the stratification, and applies the procedure to several concrete pairs of knots previously raised as open questions in the literature.

Significance. If the adaptation and termination argument hold, the work converts a non-constructive existence question into a practical, explicit algorithm with verifiable outputs. This is valuable for knot theory in non-simply-connected 3-manifolds, supplies concrete isotopies for previously unresolved cases, and extends naturally from RP^3 to general L(p,q). The constructive nature and example computations strengthen the result beyond a pure existence proof.

minor comments (3)

The abstract cites references [11] and [15] for the open-question status; the manuscript should ensure the bibliography is complete, correctly formatted, and includes full details for these and any other cited works on Hatcher's techniques or lens-space knots.
In the section describing the algorithm and its application to examples, adding a brief pseudocode outline or numbered step summary would improve readability and make the termination argument easier to follow without altering the technical content.
The discussion of lensification in general L(p,q) would benefit from one additional sentence clarifying how the stratification complexity bound scales with p and q, even if the argument is already implicit in the finite-complexity claim.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained constructive adaptation

full rationale

The paper's core contribution is an explicit algorithm obtained by adapting Hatcher's embedding-space stratification techniques to knot projectivizations in RP^3 and lens spaces L(p,q). This adaptation is presented as a direct, step-by-step constructive procedure whose termination follows from the finite complexity of the stratification; no equations, fitted parameters, or self-definitional reductions appear. The cited prior works ([11], [15]) only pose the open question and do not supply load-bearing premises for the algorithm itself. Hatcher is an external source, and the manuscript supplies independent verification via examples. The derivation therefore stands on its own without reducing to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on standard definitions of knots, embeddings, and isotopy together with the unproven assertion that Hatcher's embedding-space construction transfers directly to projectivized knots.

axioms (2)

standard math A knot is a smooth embedding of the circle S^1 into the 3-sphere S^3.
Standard definition invoked throughout knot theory and presupposed by the projectivization construction.
standard math Two embeddings are equivalent if there exists a continuous path of embeddings connecting them (an isotopy).
Definition of equivalence used to formulate the problem being solved.

pith-pipeline@v0.9.0 · 5473 in / 1349 out tokens · 78765 ms · 2026-05-08T19:17:53.597285+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear
RP^3 ≃ L(2,1) ... lensification of a knot in any lens space L(p,q)

Reference graph

Works this paper leans on

17 extracted references · 1 canonical work pages

[1]

R. Budney. A family of embedding spaces.Geometry & Topology, 13:41–83, 2008

2008
[2]

Budney and F

R. Budney and F. Cohen. On the homology of the space of knots.Geometry & Topology, 13, 2009

2009
[3]

Cattabriga and E

A. Cattabriga and E. Manfredi. Diffeomorphic vs isotopic links in lens spaces.Mediterranean Journal of Mathematics, 15:172, 2018

2018
[4]

Cattabriga, E

A. Cattabriga, E. Manfredi, and M. Mulazzani. On knots and links in lens spaces.Topology and its Applications, 160(2):430–442, 2013

2013
[5]

Y. V. Drobotukhina. An analogue of the Jones polynomial for links inRP 3 and a generalization of the Kauffman–Murasugi theorem.Algebra i Analiz, 2(3):171–191, 1990

1990
[6]

Fiedler.Polynomial One-cocycles for knots and Closed braids, volume 64 ofSeries on Knots and Everything

T. Fiedler.Polynomial One-cocycles for knots and Closed braids, volume 64 ofSeries on Knots and Everything. World Scientific, 2019

2019
[7]

R. H. Fox. Rolling.Bulletin of the American Mathematical Society, 72:162–164, 1966

1966
[8]

A. Hatcher. Topological moduli spaces of knots.https://pi.math.cornell.edu/ ~hatcher/Papers/knotspaces. pdf, 2002

2002
[9]

Havens and R

A. Havens and R. Koytcheff. Spaces of knots in the solid torus, knots in the thickened torus, and links in the 3-sphere.Geometriae Dedicata, 214:671–737, 2021

2021
[10]

Kanou and K

S. Kanou and K. Sakai. The Fox-Hatcher cycle and a Vassiliev invariant of order three.Pacific Journal of Mathematics, 323(2):281–306, 2023

2023
[11]

L. H. Kauffman, R. Mishra, and V. Narayanan. Knots inRP 3.Topology and its Applications, 377:109656, 2026

2026
[12]

Manfredi.Knots and Links in Lens Spaces: Tesi Di Dottorato

E. Manfredi.Knots and Links in Lens Spaces: Tesi Di Dottorato. PhD thesis, E. Manfredi, 2014

2014
[13]

Manolescu and M

C. Manolescu and M. Willis. A Rasmussen Invariant for Links inRP 3.Preprint. arXiv:2301.09764, 2023

work page arXiv 2023
[14]

Mishra and V

R. Mishra and V. Narayanan. Geometry of knots in real projective 3-space.Journal of Knot Theory and Its Ramifications, 32(10):2350068, 2023

2023
[15]

Narayanan

V. Narayanan. Virtual knots and projective knot homology. InProceedings of the International Conference on Knots, Quivers and Beyond (ICKQB 2025), pages 89–104. Atlantis Press, 2025

2025
[16]

Rolfsen.Knots and Links

D. Rolfsen.Knots and Links. American Mathematical Society, Providence, Rhode Island, 2003. AMS Chelsea Publishing

2003
[17]

Viro and J

O. Viro and J. Viro. Fundamental group in the projective knot theory.Topology and Geometry: A Collection of Essays Dedicated to Vladimir G. Turaev, 2021. Universidad Complutense de Madrid, Departamento de ´Algebra, Geometr ´ıa y Topolog ´ıa, Facultad de Ciencias Matem´aticas. 28040, Madrid, Spain. Email address:sergiodm@ucm.es Universidad Complutense de M...

2021