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arxiv: 2605.14185 · v1 · submitted 2026-05-13 · 🧮 math.GT

Recognition: 1 theorem link

· Lean Theorem

Taming Wild Knots with Mosaics

Mary Y. Deng , Allison K. Henrich , Sean H. Kawano , Andrew R. Tawfeek
Authors on Pith no claims yet

Pith reviewed 2026-05-15 01:36 UTC · model grok-4.3

classification 🧮 math.GT
keywords wild knotsknot mosaicsinfinite rooted treesmosaic tanglesspatial graphssingular knotsknot theory
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The pith

Infinite rooted tree mosaics represent wild knots with isolated wild points.

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

Wild knots exhibit infinite knotting behavior and have resisted standard classification techniques in topology. The paper extends Lomonaco and Kauffman's knot mosaic theory to cover a substantial subclass of these knots that feature isolated wild points. It does so by defining mosaics as infinite rooted trees, with standard mosaics placed at each vertex and embedding functions that dictate how the pieces connect in three-dimensional space. The work also introduces mosaic tangles and mosaic rigid vertex spatial graphs, treating mosaic singular knots as a special case of the latter.

Core claim

The authors establish that wild knots possessing isolated wild points can be represented by mosaics built from infinite rooted trees. Each vertex of the tree receives a finite mosaic, and embedding functions specify the spatial connections between these mosaics so that the overall object realizes a well-defined wild knot in three-space.

What carries the argument

The infinite rooted tree mosaic, which assigns mosaics to vertices and uses embedding functions to govern their connections.

If this is right

  • A substantial subclass of wild knots becomes representable by mosaic methods.
  • Mosaic tangles supply local building blocks for constructing these wild knots.
  • Mosaic rigid vertex spatial graphs, including singular knots, extend the same representation.
  • Invariants and classification tools developed for ordinary mosaics can be applied to this subclass of wild knots.

Where Pith is reading between the lines

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

  • The tree structure may generalize to wild links or other infinite embeddings once embedding consistency conditions are clarified.
  • Requiring consistent embeddings could yield new criteria distinguishing tameable wild knots from more pathological ones.
  • Mosaic representations might allow algorithmic enumeration of wild knots within the isolated-wild-point subclass.

Load-bearing premise

The wild points must be isolated and the embedding functions must be chosen consistently so that the resulting object is a well-defined knot in three-space without extra entanglements or singularities.

What would settle it

A concrete wild knot with an isolated wild point for which no assignment of mosaics to an infinite rooted tree and consistent embedding functions produces exactly that knot.

Figures

Figures reproduced from arXiv: 2605.14185 by Allison K. Henrich, Andrew R. Tawfeek, Mary Y. Deng, Sean H. Kawano.

Figure 1
Figure 1. Figure 1: A wild slip knot Because wild knots are difficult to classify, our aim is to represent a substantial subclass of them using a more discrete, combinatorial structure: knot mosaic tiles. In this paper, we develop and explore a new theory of wild mosaic knots. 1.1. History and fundamentals of wild knots. Since wild knots were defined by Fox and Artin in 1948 [FA48; Fox49], there have been some efforts devoted… view at source ↗
Figure 2
Figure 2. Figure 2: Representatives of the three classical Reidemeister moves 2. Background In this section, we review some additional important definitions and theorems, in the realms of tangles, rigid vertex spatial graphs, and knot mosaics. We will need these to develop our wild mosaic knot framework later. 1Throughout the work, “knot” may refer to either a knot or a link [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Examples of various tangles. 2.2. Rigid vertex spatial graphs. Next, we review the notions of a spatial graph and rigid vertex spatial graph. A spatial graph is a one-dimensional CW complex consisting of finitely many vertices (zero-dimensional) and finitely many (one-dimensional) edges (including loops), tamely embedded in R 3 or S 3 . Each edge is homeomorphic to a closed interval whose boundary is mappe… view at source ↗
Figure 4
Figure 4. Figure 4: Reidemeister moves IV, V* for RV4 graphs and the additional move of VI for RV2,4 graphs. 2.3. RV2,4 graphs. Throughout this paper, we will work with objects that can be viewed as rigid vertex spatial graphs having degree 4 rigid vertices, but we must also allow for degree 2 vertices for reasons that will become clear. We refer to rigid vertex spatial graphs in which all vertices have degree 2 or 4 as (2,4)… view at source ↗
Figure 5
Figure 5. Figure 5: An illustration of Reidemeister move VI and its depiction as an elementary combinatorial isotopy. Move VI, along with the moves I-IV and V* introduced in the previous section, generate PL ambient isotopy of embedded RV2,4 graphs. The proof of the RV4 case was already demonstrated by Kauffman [Kau89], so it remains to show that move VI is an elementary combinatorial isotopy. Note: it was proven in [Gra50] t… view at source ↗
Figure 6
Figure 6. Figure 6: The eleven standard mosaic tiles. Just as with ordinary knot diagrams, there is a set of Reidemeister-type moves that relate mosaics representing equivalent topological knots. This mosaic-based set of moves— shown in Figures 7, 8, and 9—is more extensive than the set of classical Reidemeister moves since planar isotopies must be exhaustively described in the mosaic setting. Moreover, natural operations tha… view at source ↗
Figure 7
Figure 7. Figure 7: The eleven mosaic planar isotopy moves [PITH_FULL_IMAGE:figures/full_fig_p006_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The mosaic Reidemeister 1 moves and Reidemeister 2 moves [PITH_FULL_IMAGE:figures/full_fig_p007_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: The mosaic Reidemeister 3 moves. The dual-colored tiles de￾note two possibilities and are in sync with their isotopic counterparts. considered to be equivalent to a knot m-mosaic if they are relatable via a sequence of mosaic Reidemeister moves and mosaic injections. Definition 2.4.2. Per [LK08], a mosaic injection ι : K(n) → K(n+1) sends a knot mosaic M(n) to M(n+1) , where M(n+1) ij = [PITH_FULL_IMAGE:f… view at source ↗
Figure 10
Figure 10. Figure 10: An injection of a tangle 3-mosaic into a tangle 4-mosaic. such as [CL13; GH20]. We will leave these fascinating avenues of study here and move on to expanding the notion of a knot mosaic to tangles. 3. Expanding Mosaics 3.1. Tangle Mosaics. Definition 3.1.1 (Tangle Mosaic). For a positive integer n, a tangle n-mosaic L is an n-mosaic in which all tiles are suitably connected and connection points are allo… view at source ↗
Figure 12
Figure 12. Figure 12: Non-examples of tangle mosaics. These are forbidden because there are two adjacent  tiles (left) or a  tile is along the boundary (right) [PITH_FULL_IMAGE:figures/full_fig_p010_12.png] view at source ↗
Figure 11
Figure 11. Figure 11: A typical RV2,4 tangle mosaic – in this case, a 7-tangle. Remark 3.2.3. Definition 3.2.1 implies that a 1 × 1 comprised of a single  is not a valid RV2,4 tangle mosaic. In other words, we require every RV2,4 tangle mosaic to contain tiles in T. If we consider the diagram of a given knot or tangle, we can use standard arguments to overlay a grid on the plane and associate a tile from T with each square to… view at source ↗
Figure 13
Figure 13. Figure 13: The new RV2,4 mosaic Reidemeister moves corresponding to IV, V*, and VI for RV2,4 graphs, accounting for the  tile. We include the pictured moves as well as their mirror images and their rotations in our expanded set of mosaic equivalences. Lemma 3.2.5 (Boundary Adjustment). Given any 1- or 2-tangle RV2,4 mosaic, there is an equivalent odd-dimensional mosaic tangle such that: • The boundaries of strands … view at source ↗
Figure 14
Figure 14. Figure 14: Boundary adjustment of a 1-tangle RV2,4 mosaic. Definition 3.2.6 (RV2,4 Tangle Mosaic Equivalence). We call two RV2,4 mosaics M and M′ boundary equivalent, denoted M ∼ b M′ , if one is a boundary adjustment of the other. Furthermore, two RV2,4 mosaic tangles M and N are RV2,4 tangle mosaic [PITH_FULL_IMAGE:figures/full_fig_p011_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: The 3-zoom map applied to an RV2,4 mosaic. For this next definition, we show how a tangle mosaic M can be inserted into another (knot or tangle) mosaic N. As the reader considers this definition, we recommend looking at the example in [PITH_FULL_IMAGE:figures/full_fig_p013_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: An example of the mosaic embedding x22 : L (5) RV × M (3) RV −→ M (15) RV . Notice that RV2,4 mosaic moves V ∗ , VIII.a, and VIII.b from [PITH_FULL_IMAGE:figures/full_fig_p014_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: The D4-action on an RV2,4 mosaic M: identity, reflection f, rotation r, and their composition rf. Definition 4.1.5 (Oriented Embeddings). For σ ∈ D4 and a mosaic embedding xij as in Definition 4.1.2, the σ-oriented embedding is the map x σ ij(M, N) = xij(σ · M, N). The fundamental embedding xij of Definition 4.1.2 is the special case σ = e, the identity element of D4. In particular, x r ij and x f ij are … view at source ↗
Figure 18
Figure 18. Figure 18: Examples of tree mosaics. (A) captures the usual notion of a knot mosaic. (B) and (C) demonstrate our ability to describe classical wild knot constructions. (D) shows that the tangles associated with each vertex need not be constant. • i : E → HomRV associates an oriented embedding of mosaics (cf. Def. 4.1.5) to every edge. We say that a tree mosaic is well-defined if, for every vertex v, there is an orde… view at source ↗
Figure 19
Figure 19. Figure 19: We can create a visual for the tree mosaic structure by draw￾ing the mosaic embeddings directly. The right-hand picture may be inter￾preted as a “top-down view” of the tree. Definition 4.2.6 (Geometric realization of a mosaic). Let M ∈ M(n) be a suitably connected n-mosaic. The geometric realization |M| ⊂ R 3 is the embedded 1-complex obtained by placing tile Mij in the region [i − 1, i] × [j − 1, j] × [0… view at source ↗
Figure 20
Figure 20. Figure 20: The geometric realization of a mosaic can be visualized as extending the mosaic tiles into cubes Definition 4.2.7 (Geometric realization of a tree mosaic). Let M = (T, F, i) be a well￾defined, suitably connected tree mosaic with root r. For each vertex v ∈ V(T), let γv = (r = v0, v1, . . . , vk = v) denote the unique path from the root to v, and let d(v) = k denote the depth of v. Define: (1) Scale factor… view at source ↗
Figure 21
Figure 21. Figure 21: Nested neighborhoods of an isolated wild point with p-index 2. Each annular region Aj,n is tame and admits a mosaic representative. Step 5: Defining the tree structure. We define the rooted tree T = (V, E) as follows: • The vertex set is V = {r} ∪ {vj,n : 1 ≤ j ≤ N, n ≥ 1}. • The root r has N children: v1,1, . . . , vN,1. • For each j and n ≥ 1, vertex vj,n has exactly one child: vj,n+1. Thus T consists o… view at source ↗
Figure 22
Figure 22. Figure 22: Ray-repeating mosaics always have one combinatorial-wild point, by construction. Remark 4.2.17. Notably, just because a knot has non-isolated wild points, that does not prohibit us from representing it as a mosaic. For example, consider a tree mosaic defined on an infinite binary tree. This has uncountably many combinatorial-wild points (i.e. infinitely many rays emanating from the root). From now on, all… view at source ↗
Figure 23
Figure 23. Figure 23: A contraction of a tree mosaic. Theorem 4.3.3. A contraction of a tree mosaic preserves the represented knot type: if M ց M′ , then |M| and |M′ | are ambient homeomorphic. Proof. Since the subtrees {Sj} are pairwise vertex-disjoint and not joined by edges, their contractions occur in disjoint regions; it is enough to prove the claim for a single subtree S. We induct on |V(S)|; the case |V(S)| = 1 is trivi… view at source ↗
Figure 24
Figure 24. Figure 24: According to the proof of Theorem 4.3.7, we apply two con￾tractions to obtain a star-like tree mosaic from an arbitrary one with finitely many combinatorial-wild points. The figures, moving from left to center, show the contraction of all finite portions of the tree to a core set of ver￾tices. Moving from center to right, we picture the contraction of a finite core set of vertices up to a depth D to the r… view at source ↗
Figure 25
Figure 25. Figure 25: Examples of rigid vertex knot and tangle mosaics with larger regions of adjacent T∞ tiles. References [Ada+20] Colin Adams et al. Encyclopedia of Knot Theory. Chapman and Hall/CRC, 2020. isbn: 1138297844. [Ada04] Colin Conrad Adams. The knot book: an elementary introduction to the math￾ematical theory of knots. American Mathematical Soc., 2004. [Ale+14] J. Alan Alewine et al. “Bounds on mosaic knots”. In:… view at source ↗
read the original abstract

Wild knots--knots with infinite knotting behavior--have resisted traditional methods of knot classification, making them more of a curiosity in topology than a subject of sustained investigation. In this paper, we present a new way to investigate these objects. We extend Lomonaco and Kauffman's knot mosaic theory to represent a substantial subclass of wild knots that have isolated wild points. Our mosaics consist of infinite rooted trees with mosaics assigned to vertices and embedding functions governing connections. In developing this framework, we also introduce a notion of mosaic tangles as well as mosaic rigid vertex spatial graphs of which mosaic singular knots are a special case.

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.

Referee Report

2 major / 2 minor

Summary. The paper extends Lomonaco and Kauffman's knot mosaic theory to a subclass of wild knots possessing isolated wild points. The central construction represents such knots via infinite rooted trees, with a mosaic assigned to each vertex and embedding functions controlling the connections between them; the framework also introduces mosaic tangles and mosaic rigid vertex spatial graphs (of which mosaic singular knots are a special case).

Significance. If the embedding functions can be shown to produce globally consistent embeddings whose only wild points are the prescribed isolated ones, the framework would supply a combinatorial language for a previously intractable class of objects and could support the development of new invariants or enumeration techniques for wild knots.

major comments (2)
  1. [Definition of embedding functions and the infinite assembly (likely §3–4)] The central claim that the infinite-tree construction yields a well-defined embedding in 3-space with precisely the prescribed isolated wild points rests on the existence and consistency of the embedding functions. The manuscript must supply an explicit verification (or at least a non-vacuous existence argument) that these functions can be chosen so that the limit map remains injective and proper, with no unintended accumulation of crossings or singularities at connection loci or at infinity.
  2. [Introduction and main construction] The paper asserts that the construction captures a 'substantial subclass' of wild knots with isolated wild points, yet provides neither a precise characterization of this subclass nor concrete examples (e.g., the wild trefoil or Fox-Artin wild knots) together with explicit mosaic assignments and embedding functions that realize them.
minor comments (2)
  1. [Definitions] Notation for the infinite rooted tree and the assignment of mosaics to vertices should be made fully explicit (e.g., a formal tuple or diagram) to avoid ambiguity when the tree is infinite.
  2. [Mosaic tangles] The relationship between the new mosaic tangles and classical tangle theory should be stated more clearly, including whether every classical tangle arises as a finite-depth mosaic tangle.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback on our manuscript. We address each major comment below and describe the revisions we will make to strengthen the paper.

read point-by-point responses

  1. Referee: [Definition of embedding functions and the infinite assembly (likely §3–4)] The central claim that the infinite-tree construction yields a well-defined embedding in 3-space with precisely the prescribed isolated wild points rests on the existence and consistency of the embedding functions. The manuscript must supply an explicit verification (or at least a non-vacuous existence argument) that these functions can be chosen so that the limit map remains injective and proper, with no unintended accumulation of crossings or singularities at connection loci or at infinity.

    Authors: We agree that an explicit verification is required. In the revised manuscript we will insert a new subsection (in §4) that constructs the embedding functions inductively over finite subtrees. We prove that, by choosing the functions to be homeomorphisms on the attaching spheres and to contract diameters sufficiently rapidly (with a uniform rate independent of the tree depth), the pointwise limit is a continuous, injective, proper map into R^3 whose only non-tame points are the prescribed isolated wild points. The argument rules out unintended accumulation by controlling the diameters of the images of the connection loci. revision: yes

  2. Referee: [Introduction and main construction] The paper asserts that the construction captures a 'substantial subclass' of wild knots with isolated wild points, yet provides neither a precise characterization of this subclass nor concrete examples (e.g., the wild trefoil or Fox-Artin wild knots) together with explicit mosaic assignments and embedding functions that realize them.

    Authors: We accept that concrete examples and a clearer delineation of the subclass are needed. In the revision we will add an examples subsection that explicitly realizes the Fox-Artin wild knot (and the wild trefoil) by specifying the infinite rooted tree, the local mosaic assigned to each vertex, and the embedding functions on the attaching intervals. We will also refine the introduction to characterize the subclass as those wild knots whose wild points are isolated and admit a tree decomposition into tame arcs whose diameters decrease rapidly enough to allow the mosaic assembly to converge. revision: yes

Circularity Check

0 steps flagged

No circularity: new mosaic construction for wild knots is self-contained

full rationale

The paper introduces a direct definitional extension of Lomonaco-Kauffman mosaics to infinite rooted trees equipped with vertex mosaics and embedding functions for wild knots with isolated wild points. No equations reduce any claimed prediction or result to a fitted parameter or self-referential input; the framework is built by explicit construction rather than by renaming or deriving from prior fitted quantities within the paper. The cited mosaic theory is external prior work, and the new objects (mosaic tangles, rigid vertex graphs) are defined outright without load-bearing self-citation chains or ansatzes smuggled from the authors' own earlier results. The construction stands as an independent definitional framework.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 2 invented entities

The framework rests on the standard axioms of knot mosaics from Lomonaco-Kauffman plus the new stipulation that wild points are isolated and that embedding functions exist to glue the local mosaics into a global wild knot. No free parameters or invented physical entities are introduced.

axioms (1)
  • domain assumption Knot mosaics are well-defined combinatorial objects that can be assigned to vertices of a tree and connected via embedding functions without creating extraneous singularities.
    Invoked when the paper states that mosaics are assigned to vertices and embedding functions govern connections.
invented entities (2)
  • mosaic tangle no independent evidence
    purpose: To generalize the mosaic framework to tangles within the wild-knot setting.
    Introduced as part of the new machinery alongside the main wild-knot mosaics.
  • mosaic rigid vertex spatial graph no independent evidence
    purpose: To handle singular knots and graphs with rigid vertices in the mosaic language.
    Defined as a special case within the same tree-based mosaic system.

pith-pipeline@v0.9.0 · 5407 in / 1446 out tokens · 27732 ms · 2026-05-15T01:36:32.689215+00:00 · methodology

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Reference graph

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