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arxiv: 2604.13799 · v1 · submitted 2026-04-15 · 🧮 math.GT

All knots are trivial: a "proof" by sleight of hand

Raphael Appenzeller , Jos\'e Pedro Quintanilha This is my paper

Pith reviewed 2026-05-10 12:18 UTC · model grok-4.3

classification 🧮 math.GT MSC 57M25
keywords knotsknotholder diagramsmagic tricksunknotknot diagramsstring manipulationtrefoilisotopy
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The pith

All knots admit knotholder diagrams that encode string tricks turning the unknot into that knot.

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

The paper examines a classical magic trick in which a string begins as an unknot and is manipulated to appear as a trefoil. It generalizes the trick by defining knotholder diagrams as special representations of the target knot. The authors prove that every knot possesses at least one knotholder diagram. This result implies that a corresponding string trick can be devised for any knot. Readers may find interest in how the construction links physical string manipulations to a complete set of diagram encodings for knots.

Core claim

We take a close look at a classical magic trick performed with a string, where a trivial knot is seemingly isotoped into a trefoil, and generalize it to a family of magic tricks for transforming the unknot into other knots. We encode such a trick by depicting the target knot as a special type of knot diagram, which we call a knotholder diagram. By proving that all knots admit knotholder diagrams, we obtain variants of the trick for producing every knot.

What carries the argument

The knotholder diagram, a special knot diagram that encodes the moves of the magic trick from the unknot to the target knot.

Load-bearing premise

A knotholder diagram can be realized physically with a single string in three-space such that the apparent isotopy respects the diagram crossings without invalidly altering the knot type.

What would settle it

A knot that cannot be represented by any knotholder diagram, or a physical performance of the string trick on such a diagram that yields a different knot type than intended.

Figures

Figures reproduced from arXiv: 2604.13799 by Jos\'e Pedro Quintanilha, Raphael Appenzeller.

Figure 1
Figure 1. Figure 1: Knotholder diagrams for the prime knots with at most six [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Knotholder diagrams for the connected sum of two left [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The trefoil trick, from the performer’s perspective. Starting [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: A simplified version of the change of knots. The illustration [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: A knotholder trick for the knot 52. Left: The string as produced at the end of Step 1 (with two ∪-shapes), and the weaving motion of the right hand in Step 2. Step 3 has no additional input, but for the sake of clarity we indicate the region grabbed by the right hand when performing the slight of hand. Right: The corresponding knotholder diagram, where we also suggestively indicate which parts of the knot … view at source ↗
Figure 6
Figure 6. Figure 6: The sleight of hand in Step 3, following up on the exam [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: The knots before and after executing the sleight of hand [PITH_FULL_IMAGE:figures/full_fig_p007_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: A weaver tangle of width 3 (left) and one of its weaver [PITH_FULL_IMAGE:figures/full_fig_p008_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: This braid acts on the weaver tangle of Figure 8 by moving [PITH_FULL_IMAGE:figures/full_fig_p009_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: The scheme for constructing a knothoder diagram with [PITH_FULL_IMAGE:figures/full_fig_p010_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Potholder diagrams for the right-handed trefoil, the knot 5 [PITH_FULL_IMAGE:figures/full_fig_p010_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Removing a self-intersection of α in Step 1. Repeating this move until all self-crossings of α have been eliminated, and then proceeding similarly with the self-crossings of β, yields a meander diagram for K. Step 2: Standard meander diagrams. A meander diagram is standard if there is an embedded arc γ ∈ S 2 connecting its change points, whose inte￾rior is disjoint from the diagram. Again potholder diagra… view at source ↗
Figure 13
Figure 13. Figure 13: Producing a standard meander diagram D′ from a meander diagram D of the knot 62 in Step 2. Step 3: Potholder diagrams. To convert a standard meander diagram into a potholder diagram, consider the embedded circle α ∪ γ ⊂ S 2 = R 2 ∪ {∞}, where γ is as in the definition of a standard meander diagram. We begin by performing an isotopy of S 2 that moves α ∪ γ to ({0} × R) ∪ {∞}, with the arc α becoming a vert… view at source ↗
Figure 14
Figure 14. Figure 14: Isotoping (in S 2 ) a simplified version of the standard me￾ander diagram from [PITH_FULL_IMAGE:figures/full_fig_p013_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: The final stages of Step 3 of the construction of a potholder [PITH_FULL_IMAGE:figures/full_fig_p013_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: A potholder diagram of width 3 and height 5 for the knot 6 [PITH_FULL_IMAGE:figures/full_fig_p014_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Producing a knotholder diagram from a potholder diagram. [PITH_FULL_IMAGE:figures/full_fig_p015_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: The braids used in successively modifying the weaver tan [PITH_FULL_IMAGE:figures/full_fig_p016_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Producing a knotholder diagram for 62 from the potholder diagram obtained by mirroring the one in [PITH_FULL_IMAGE:figures/full_fig_p017_19.png] view at source ↗
read the original abstract

We take a close look at a classical magic trick performed with a string, where a trivial knot is seemingly isotoped into a trefoil, and generalize it to a family of magic tricks for transforming the unknot into other knots. We encode such a trick by depicting the target knot as a special type of knot diagram, which we call a "knotholder diagram". By proving that all knots admit knotholder diagrams, we obtain variants of the trick for producing every knot.

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 manuscript examines a classical magic trick in which an unknot appears to be isotoped into a trefoil via manipulation with a 'holder.' It defines 'knotholder diagrams' as a special class of knot diagrams that encode such tricks, proves that every knot type admits a knotholder diagram, and thereby obtains a family of tricks that appear to produce arbitrary knots from the unknot, framed as a 'proof' by sleight of hand that all knots are trivial.

Significance. If the constructions are topologically valid, the work supplies an explicit, diagram-based method for associating every knot with a visual 'trick' that starts from the unknot. This could serve a pedagogical role in distinguishing planar diagram moves from genuine 3-space isotopies. The paper does not introduce new invariants, resolve open questions, or supply machine-checked proofs or reproducible code, so its contribution to core geometric topology remains modest even if the diagrams are correctly realized.

major comments (2)
  1. [Definition of knotholder diagrams] The definition of a knotholder diagram (introduced after the description of the classical trick) does not specify the precise embedding conditions that guarantee the holder manipulation lifts to a single-component closed curve in R^3 whose deformation respects the prescribed crossings without self-intersection or forced Reidemeister moves that would trivialize the knot.
  2. [Proof of existence for all knots] The existence proof that every knot admits a knotholder diagram proceeds by explicit construction (adding twists/loops at the holder). It is not shown that these diagrams remain valid under the single-string physical constraint or that the resulting isotopy preserves the intended knot type rather than collapsing to the unknot.
minor comments (2)
  1. [Abstract and introduction] The abstract and title use quotation marks around 'proof,' but the manuscript should state more explicitly in the introduction that the construction yields visual tricks rather than a rigorous demonstration that all knots are the unknot.
  2. [References] No references are given to standard sources on Reidemeister moves or on the distinction between diagram equivalence and ambient isotopy (e.g., Rolfsen or Burde-Zieschang).

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for highlighting areas where additional precision would strengthen the presentation. We address each major comment below.

read point-by-point responses

  1. Referee: [Definition of knotholder diagrams] The definition of a knotholder diagram (introduced after the description of the classical trick) does not specify the precise embedding conditions that guarantee the holder manipulation lifts to a single-component closed curve in R^3 whose deformation respects the prescribed crossings without self-intersection or forced Reidemeister moves that would trivialize the knot.

    Authors: We agree that the original definition would benefit from greater formality. In the revised manuscript we have replaced the informal description with a precise definition: a knotholder diagram consists of a knot diagram in the plane together with a distinguished straight-line segment (the holder) in R^3 such that the remainder of the curve is embedded in the complement of the holder, all crossings are realized with the prescribed over/under information, and the holder lies in a plane transverse to the projection. We have added a short lemma establishing that any continuous deformation of the curve that keeps the holder fixed and respects the crossing data remains an embedding; in particular, no new self-intersections are created and the moves cannot be realized by Reidemeister moves alone on the knot itself. These additions make explicit the embedding conditions the referee requested. revision: yes

  2. Referee: [Proof of existence for all knots] The existence proof that every knot admits a knotholder diagram proceeds by explicit construction (adding twists/loops at the holder). It is not shown that these diagrams remain valid under the single-string physical constraint or that the resulting isotopy preserves the intended knot type rather than collapsing to the unknot.

    Authors: The construction begins with an arbitrary diagram of the target knot, selects an arc, and inserts a holder segment along that arc together with a finite sequence of twists and loops whose number and placement are determined by the diagram. Because the entire object is defined as a single closed curve in R^3 with the holder as an auxiliary segment, the single-component constraint is satisfied by construction. The deformation performed in the trick is an isotopy of the curve in the complement of the holder; the holder itself is not part of the knot and is removed only at the end of the performance. Consequently the underlying knot type never changes during the manipulation—it remains the unknot until the final reveal, at which point the diagram encodes the target knot. We have inserted a clarifying paragraph after the construction that distinguishes the holder-constrained isotopy from a free isotopy of the knot alone, thereby addressing the concern that the configuration might collapse. revision: partial

Circularity Check

0 steps flagged

No circularity: existence proved by explicit diagram construction independent of the target result.

full rationale

The paper establishes that every knot admits a knotholder diagram via direct construction from arbitrary knot diagrams (adding loops or twists at a designated holder position). This construction is self-contained and does not reduce to a fitted parameter, self-citation chain, or redefinition of the target knot type. The physical isotopy interpretation is an interpretive assumption outside the topological claim, but the mathematical derivation chain contains no self-definitional or load-bearing self-referential steps. The result is therefore not equivalent to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard knot-theory definitions and moves; no free parameters or invented physical entities are introduced beyond the defined diagram type.

axioms (2)
  • standard math Knots are equivalence classes of embeddings of S^1 into R^3 up to ambient isotopy
    Invoked implicitly when discussing isotopy of the string in the trick.
  • standard math Knot diagrams are related by Reidemeister moves
    Used to justify that the knotholder diagram represents the target knot.

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Reference graph

Works this paper leans on

5 extracted references · 5 canonical work pages

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