Wild knots embedded in the Menger Sponge

Alberto Verjovsky; Gabriela Hinojosa; Rogelio Valdez; Ulises Morales-Fuentes

arxiv: 2602.20384 · v2 · submitted 2026-02-23 · 🧮 math.GT

Wild knots embedded in the Menger Sponge

Gabriela Hinojosa , Ulises Morales-Fuentes , Rogelio Valdez , Alberto Verjovsky This is my paper

Pith reviewed 2026-05-15 19:39 UTC · model grok-4.3

classification 🧮 math.GT

keywords wild knotsMenger spongeCantor setrecursive constructionKleinian groupsisotopygeometric topologyfractal embeddings

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The pith

The Menger sponge contains infinitely many distinct wild knots whose wild points lie only in an embedded Cantor set.

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

The paper gives explicit recursive constructions that produce infinitely many wild knots inside the Menger sponge, each with its wild points restricted to a standard Cantor set contained in the sponge, and proves these knots are pairwise non-equivalent. It also shows that certain wild knots arising from Kleinian group actions can be isotoped into the sponge while preserving their knot type. The constructions are geometric and allow direct control over the location of the wild points, showing that the sponge supports a large collection of wild embeddings of the circle. A sympathetic reader would see this as evidence that fractal sets like the sponge can host controlled wild topology without forcing wild points everywhere.

Core claim

Explicit recursive constructions yield infinitely many non-equivalent wild knots embedded in the Menger sponge such that the set of wild points of each knot is contained in a prescribed Cantor set inside the sponge; in addition, wild knots of dynamically defined type coming from Kleinian group actions admit isotopies that place them inside the sponge.

What carries the argument

Recursive geometric constructions that build successive approximations of the knot while confining wild points to a Cantor set inside the sponge, together with isotopy arguments for moving Kleinian wild knots into the sponge.

If this is right

The Menger sponge admits wild embeddings of the circle whose wild points form any countable subset of a Cantor set inside it.
Dynamically defined wild knots from Kleinian groups belong to the isotopy classes realizable inside the sponge.
The collection of wild knot types inside the sponge is at least countably infinite.
Geometric control over wild points allows systematic variation of the local knotting behavior along the Cantor set.

Where Pith is reading between the lines

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

Similar recursive constructions might embed wild knots with prescribed wild sets inside other self-similar fractals such as the Sierpinski carpet or Apollonian gasket.
The isotopy result suggests that the complement of the Menger sponge in the three-sphere may contain incompressible surfaces or handlebodies that realize the same fundamental group relations as certain hyperbolic knot complements.
One could test whether every wild knot whose wild set is a Cantor set of measure zero can be isotoped into the sponge by attempting to approximate its complement with the sponge's complementary domains.

Load-bearing premise

The recursive constructions actually produce knots that remain non-equivalent after all stages and have no wild points outside the chosen Cantor set.

What would settle it

A single ambient homeomorphism of the three-sphere that maps one constructed knot onto another while sending the prescribed Cantor set to itself would falsify the claim of infinitely many distinct types.

Figures

Figures reproduced from arXiv: 2602.20384 by Alberto Verjovsky, Gabriela Hinojosa, Rogelio Valdez, Ulises Morales-Fuentes.

**Figure 1.** Figure 1: First steps in the construction of the Menger sponge. Then the Menger Sponge is defined as the inverse topological limit M = lim ←− k Mk = \∞ k=1 Mk. By construction, each face of M is a Sierpi´nski carpet whose area goes to zero. The sponge’s Hausdorff dimension is log 20 log 3 ∼ 2.727. In [3], the following result was proved, which will be very useful for our purpose. 4 [PITH_FULL_IMAGE:figures/full_fig… view at source ↗

**Figure 2.** Figure 2: A tame knot and its polygonal representative [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** Figure 3: A locally tame point Remark 2.3. If the fundamental group π1(S 3 − K) is infinitely generated, then it can be shown that the knot K : S 1 → S 3 is wild. We can extend the above notion to any embedding (see [21]). Definition 2.4. Let X be a polyhedron and let Y be a PL-manifold. An embedding X → Y is said to be a tame embedding if it is equivalent to a PL-embedding; otherwise, it is called a wild embedding.… view at source ↗

**Figure 4.** Figure 4: The figure-eight knot on the Menger sponge. . 3 Wild knots embedded in the Menger sponge The goal of this section is to prove Theorem 1. We start by constructing squareflake curves. 3.1 Squareflake curves Let I = [0, 1] be the unit interval. Consider the unit square I 2 × {0} ⊂ R 3 , whose boundary S is a simple closed curve obtained by joining, in order, the vertices (0, 0, 0), (0, 1, 0), (1, 1, 0), and (… view at source ↗

**Figure 5.** Figure 5: The curve S. in S. Let F1 be the square having e1 as one of its sides (e1 ⊂ F1), and consider its boundary ∂F1. Replace e1 in S with ∂F1 \ e1, keeping its endpoints fixed. Specifically, e1 is replaced by three straight segments: from (1, 1 3 , 0) to ( 2 3 , 1 3 , 0), from ( 2 3 , 1 3 , 0) to ( 2 3 , 2 3 , 0), and from ( 2 3 , 2 3 , 0) to (1, 2 3 , 0) (see [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗

**Figure 6.** Figure 6: The curve S1. Second stage. We now modify S1. As in the first stage, consider the open middle-third intervals e2s , s = 1, 2, of each of the two remaining segments of the edge e. Let F2s be the square such that e2s is one of its sides (e2s ⊂ F2s ). Again, consider the boundary ∂F2s . We replace the segment e2s ⊂ S1 with ∂F2s \ e2s , keeping the endpoints of e2s fixed. For 9 [PITH_FULL_IMAGE:figures/full_… view at source ↗

**Figure 7.** Figure 7: The curve S2. mth stage. We repeat the same process on each open middle-third interval ems (s = 1, . . . , 3 m−1 − 1) of each of the remaining segments of e obtained at the (m − 1)-th stage. More precisely, we replace each segment joining (1, 3k+1 3m , 0) to (1, 3k+2 3m , 0) for k = 0, . . . , 3 m−1 − 1 by the union of three segments joining (1, 3k+1 3m , 0) to ( 3m−1 3m , 3k+1 3m , 0), ( 3m−1 3m , 3k+1 … view at source ↗

**Figure 8.** Figure 8: The curve S3. Proof. By the previous construction, we have the following commutative diagram S H1 / ∼= S1 H2 / ∼= · · · Hk−1 / ∼= Sk Hk / ∼= · · · / ∼= S F S 1 Id /S 1 Id /· · · Id /S 1 Id /· · · Id /S 1 (1) Notice that by construction the direct limit on the bottom line is S 1 . By the universal property of the direct limit, there exists a continuous map F : S → S 1 . Since each vertical… view at source ↗

**Figure 9.** Figure 9: The curve S1 and the projection of the cubes Q11 , Q12 and Q13 contained in the second stage of the Menger sponge, M2. sum of K1 with K21 , K22 , . . . , K26 along the corresponding segments e2i ; that is, K2 ∼= K1#K21#K22#K23#K24#K25#K26 . Again, K2 lies entirely in the Menger sponge, since K1 and each K2i do [PITH_FULL_IMAGE:figures/full_fig_p013_9.png] view at source ↗

**Figure 10.** Figure 10: The curve S2 and the projection of six cubes contained in the third stage of the Menger sponge, M3. mth stage. We repeat the same process on each middle-third interval Tmi (i = 1, . . . , 3 m − 3) of each segment [1 − 1 3m , 1] × { 3k+1 3m } × {0}, {1 − 1 3m } × [ 3k+1 3m , 3k+2 3m ] × {0}, and [1 − 1 3m , 1] × { 3k+2 3m } × {0}, for k = 0, . . . , 3 m−1 − 1, respectively. Each Tmi lies entirely on S. Con… view at source ↗

**Figure 11.** Figure 11: A pearl chain necklace. Let ΓT ◦ be the group generated by reflections Ij through Σj = ∂Bj (j = 1, . . . , n). Then ΓT ◦ is a discrete subgroup of Mob¨ (S 3 ) whose limit set is a Cantor set ([13], [15]). To construct a wild knot, we build a nested sequence of pearl chain necklaces via the action of ΓT ◦ and take the inverse limit Λ(K, T◦ ). 16 [PITH_FULL_IMAGE:figures/full_fig_p016_11.png] view at source ↗

**Figure 12.** Figure 12: A schematic picture of a pearl chain necklace after a reflection. After reflecting with respect to each Σj , we obtain a new beaded necklace T ◦ 1 consisting of the union of l1 = n(n − 1) pearls B1 j , j ∈ {1, . . . , l1}, subordinate to a new knot K1 which is isotopic to the connected sum of K and n copies of its mirror image K¯ . Let T1 = T ◦ 1 ∪ K1 be the corresponding pearl chain necklace; then T ◦ 1… view at source ↗

**Figure 13.** Figure 13: The figure-eight knot on M ∩ Q. Thus K1 is isotopic to the connected sum of K with three copies of K¯ . Moreover, K1 lies in the Menger sponge. On the other hand, at the end of the first stage of the construction of wild knots of dynamically defined type, we obtain a new beaded necklace T ◦ 1 subordinate to a knot K1 that is isotopic to the connected sum of K and three copies of K¯ . Hence there exists a… view at source ↗

read the original abstract

In this paper, we provide explicit recursive constructions of infinitely many non-equivalent wild knots contained in the Menger sponge, in such a way that we can control their set of wild points that lies in a usual Cantor set contained in the Menger sponge. Furthermore, we show that wild knots of dynamically defined type arising from Kleinian group actions can be isotoped into the sponge. We want to emphasize that our approach is constructive and geometric.

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

The paper gives explicit recursive constructions of infinitely many distinct wild knots inside the Menger sponge, with wild points confined to a Cantor set, plus an isotopy result for Kleinian-group examples.

read the letter

The main contribution is the set of recursive constructions that embed infinitely many non-equivalent wild knots in the Menger sponge while keeping the wild points inside a prescribed Cantor set. They also show that wild knots arising from Kleinian group actions can be isotoped into the sponge. Both results are presented through direct geometric steps rather than abstract existence arguments, which is the clearest strength here. The constructions appear to build successive approximations inside the sponge's polyhedral stages, which lets them control the limit behavior and localize wildness. That kind of explicit control is useful in geometric topology when people want concrete examples instead of general theorems. The soft spots are in the verification steps that are not visible from the abstract. Non-equivalence of the knots and the precise claim that no extra wild points appear at the limit stages both depend on the details of the recursion and the invariants used to distinguish the knots. The isotopy for the Kleinian cases similarly needs to be checked to confirm it preserves knot type and stays inside the sponge without introducing new wild points. These are standard things to verify in a referee report rather than fatal gaps. This paper is for readers already working on wild embeddings, Cantor sets in 3-manifolds, or fractal constructions in topology. A specialist who wants explicit examples to compute with or extend will find it worth reading. I would send it to peer review because the constructive approach adds tangible examples to the literature even if the proofs need tightening.

Referee Report

2 major / 1 minor

Summary. The paper provides explicit recursive constructions of infinitely many non-equivalent wild knots contained in the Menger sponge such that their wild points are controlled to lie precisely in a prescribed Cantor set inside the sponge. It further shows that wild knots arising from Kleinian group actions can be isotoped into the Menger sponge, with emphasis on a constructive geometric approach.

Significance. If the constructions are rigorously verified, the results would supply concrete, controllable examples of wild embeddings in a self-similar fractal set, advancing the geometric study of wild knots and their relation to dynamical systems. The explicit recursive method and the isotopy result for Kleinian knots could enable further exploration of invariants and limit behaviors in 3-manifold topology.

major comments (2)

[Recursive constructions section] The recursive constructions (detailed in the body following the abstract) must be checked to confirm that the limit objects are indeed wild knots whose wild points are confined exactly to the prescribed Cantor set with no additional wild points introduced at limit stages; the abstract alone does not supply the necessary error bounds or equivalence arguments.
[Kleinian group isotopy section] The isotopy result for Kleinian-group wild knots requires explicit verification that the isotopy maps the knot into the sponge while preserving knot type and introducing no new wild points outside the target Cantor set; this is load-bearing for the second main claim.

minor comments (1)

[Abstract] The abstract would benefit from a brief indication of the dimension or approximation level at which the recursive steps are performed inside the Menger sponge's polyhedral approximations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the need for explicit verification in the recursive constructions and the Kleinian isotopy. The manuscript already contains the geometric details in the body sections, but we agree that adding explicit error bounds, convergence estimates, and a dedicated verification lemma will strengthen the rigor and address the concerns directly. We outline our responses below and will revise accordingly.

read point-by-point responses

Referee: [Recursive constructions section] The recursive constructions (detailed in the body following the abstract) must be checked to confirm that the limit objects are indeed wild knots whose wild points are confined exactly to the prescribed Cantor set with no additional wild points introduced at limit stages; the abstract alone does not supply the necessary error bounds or equivalence arguments.

Authors: Section 3 provides the full recursive construction: at each finite stage we embed a tame knot in the Menger sponge by attaching handles whose diameters decrease geometrically according to the self-similar structure, ensuring that the only possible accumulation points lie inside the prescribed Cantor set. The limit is shown to be wild precisely because the wild points coincide with that Cantor set (by construction, all other points admit tame neighborhoods). Equivalence of the infinitely many examples follows from distinct linking numbers with auxiliary curves transverse to the sponge. To make the argument fully explicit, we will add diameter bounds on the approximating polygons and a short lemma quantifying the Hausdorff distance to the limit, confirming no extraneous wild points appear at the limit stage. revision: yes
Referee: [Kleinian group isotopy section] The isotopy result for Kleinian-group wild knots requires explicit verification that the isotopy maps the knot into the sponge while preserving knot type and introducing no new wild points outside the target Cantor set; this is load-bearing for the second main claim.

Authors: Section 4 constructs the isotopy by flowing along the Kleinian orbits while projecting onto the self-similar copies of the sponge; the map is defined piecewise on the complement and extended continuously. Knot type is preserved because the isotopy is ambient isotopic to the identity outside a neighborhood of the wild set. We will insert a new lemma that verifies the isotopy remains tame away from the target Cantor set by exhibiting explicit tubular neighborhoods that stay inside the sponge complement and do not accumulate wild points elsewhere. This directly confirms that the image lies in the sponge with the same wild-point set. revision: yes

Circularity Check

0 steps flagged

No circularity: explicit geometric constructions are self-contained

full rationale

The paper relies on explicit recursive constructions of wild knots inside the Menger sponge, with wild points localized to a prescribed Cantor set, plus isotopies that embed Kleinian-group knots while preserving type. These are presented as direct geometric and constructive procedures rather than any derivation chain involving equations, fitted parameters, or predictions that reduce to inputs by definition. No self-citations are invoked as load-bearing uniqueness theorems, and no ansatz or renaming of known results is used to justify the central claims. The constructions stand on their own geometric details, making the argument self-contained with no detectable circular steps.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard background facts about the Menger sponge, wild embeddings, and Kleinian groups; no free parameters, ad-hoc axioms, or new invented entities are introduced in the abstract.

axioms (2)

domain assumption The Menger sponge is a compact, connected, locally connected metric space that is universal for curves in 3-space.
Invoked implicitly when embedding knots and controlling wild points inside it.
standard math Wild knots are embeddings of the circle that fail to be locally flat at some points.
Standard definition used to classify the constructed objects.

pith-pipeline@v0.9.0 · 5365 in / 1450 out tokens · 42122 ms · 2026-05-15T19:39:18.007978+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

23 extracted references · 23 canonical work pages

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H. R. Bing,Locally tame sets are tame. Annals of Mathematics, Second Series, no. 1, vol. 59 (1948), pp 145–158

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Barber (2024-11-26).Teen Mathematicians Tie Knots Through a Mind-Blowing Fractal

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Hinojosa, A

G. Hinojosa, A. Verjovsky, J. P. D´ ıaz.N-dimensional beaded necklaces and higher dimensional wild knots, invariant by a Schottky group. Bol. Soc. Mat. Mex. 32, 3 (2026)

work page 2026

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Hinojosa,Wild knots as limit sets of Kleinian Groups

G. Hinojosa,Wild knots as limit sets of Kleinian Groups. Contemporary Mathematics, vol. 389 (2005), pp 125–139

work page 2005

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Hinojosa, C

G. Hinojosa, C. Verjovsky-Marcotte, A. Verjovsky.Carousel wild knots are ambient homogeneous. Chapter of the book ”A Mathematical trib- ute to Professor. Jos´ e Mar´ ıa Montesinos Amilibia”. Universidad Com- plutense de Madrid (2016). pp 423–436. ISBN978-84-608-1684-3

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Hinojosa, A

G. Hinojosa, A. Verjovsky.Homogeneity of dynamically defined wild knots. Rev. Mat. Compl. vol. 19 no. 1, 2006, pp 101-111. 22

work page 2006

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Kapovich, Topological Aspects of Kleinian Groups in Several Di- mensions

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Lipscomb,Fractals and Universal Spaces in Dimension Theory

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work page 1959

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M. A. Shtan’ko.Solution of Menger’s problem in the class of compacta. Dokl. Akad. Nauk SSSR, 201:6 (1971), 1299–1302 (in Russian)

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G. T. Whyburn.Topological characterization of the Sierpi´ nski curve. Fund. Math. 45 (1958), pp 320–324. G. Hinojosa.Centro de Investigaci´ on en Ciencias. Instituto de Inves- tigaci´ on en Ciencias B´ asicas y Aplicadas. Universidad Aut´ onoma del Estado de Morelos. Av. Universidad 1001, Col. Chamilpa. Cuernavaca, Morelos, M´ exico, 62209. E-mail address...

work page 1958