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arxiv: 2606.24268 · v1 · pith:EPXU6JSEnew · submitted 2026-06-23 · 🧮 math.GT

The 4-move kills the Alexander polynomial

Nikos Askitas This is my paper

Pith reviewed 2026-06-25 22:03 UTC · model grok-4.3

classification 🧮 math.GT
keywords 4-moveAlexander polynomialknot theoryunknotting operationknot invariantstangle movesisotopy
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The pith

Every knot can be reduced to one with trivial Alexander polynomial via 4-moves and isotopies.

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

The paper proves that for any knot there exists a finite sequence of 4-moves combined with isotopies that produces a knot whose Alexander polynomial equals the constant 1. This is established while leaving open the separate question of whether 4-moves alone can unknot every knot. The result matters because the Alexander polynomial distinguishes many knots and the demonstration shows that this distinction can be removed by the given local operation. A sympathetic reader cares because the finding narrows what must still be checked to decide if the 4-move unknots all knots.

Core claim

The author shows that every knot can be transformed, by a finite sequence of 4-moves and isotopies, into a knot whose Alexander polynomial is the constant polynomial 1.

What carries the argument

The 4-move, a local replacement of one four-strand tangle configuration by another in a knot diagram.

If this is right

  • The Alexander polynomial imposes no obstruction to reducing a knot under 4-moves to the trivial case.
  • The open question whether 4-moves unknot all knots now reduces to the subclass of knots that already have trivial Alexander polynomial.
  • Any knot is connected by 4-moves and isotopy to at least one knot with trivial Alexander polynomial.
  • The 4-move can be used to simplify knots with respect to the Alexander polynomial without necessarily producing the unknot.

Where Pith is reading between the lines

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

  • If 4-moves are later shown to unknot every knot, the first stage of any such unknotting sequence can be chosen to make the Alexander polynomial trivial.
  • Explicit sequences of 4-moves could be computed for small-crossing knots to exhibit the reduction in practice.
  • The same reduction technique might be tested on other polynomial invariants that change under the 4-move.
  • The result separates the Alexander polynomial from the question of 4-move equivalence classes of knots.

Load-bearing premise

The 4-move alters the Alexander polynomial in a controlled manner that permits repeated application until the polynomial becomes trivial.

What would settle it

A knot for which every possible sequence of 4-moves and isotopies produces a knot whose Alexander polynomial is never the constant 1.

Figures

Figures reproduced from arXiv: 2606.24268 by Nikos Askitas.

Figure 1
Figure 1. Figure 1: The 4-move [1]. In short the conjecture, posted in 1994 by J. Przytycki as Conjecture 2.2 (a) in Rob Kirby’s list of Problems in Low-Dimensional Topology, is known to hold for many knots including knots of up to 12 crossings. For many years the 2-cable of the trefoil was the smallest potential counterexample to the ∗ I started this paper in October of 2020 as a response to the Covid-19 lockdown imposed at … view at source ↗
Figure 2
Figure 2. Figure 2: The map k maps n-graph projections (left) to knot projections (right) taking into account arc framing and hook sign: a contained graph (top) and a non-contained one (bottom). The degrees of freedom available to us when we create knot projections from n-graphs involve the number of edges of the graph, the placement of the attaching feet of these edges on the circle S (the ”attaching data”), the entanglement… view at source ↗
Figure 3
Figure 3. Figure 3: A band’s b under-crossing of S becomes an overcrossing at the expense of adding two new bands attached to it which can then be slid onto S along b. Now slide the feet of these newly introduced bands along the band b towards its foot and onto S. If we do this in order of appearance of under￾crossings of b as we traverse it from its foot towards its hook then the newly introduced bands will move parallel to … view at source ↗
Figure 4
Figure 4. Figure 4: We can change the sign of a clasp by a 4-move and we can add [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The 4-move applied near the feet of bands [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Arrow 1 is simply isotopy. Arrow 2 uses the hook of the right [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: By using 4-moves and isotopies we can add a full twist between [PITH_FULL_IMAGE:figures/full_fig_p007_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The 4-move induces a 4-move on bands Proof. The proof follows by observing that the sign of the full twist on the right hand side of [PITH_FULL_IMAGE:figures/full_fig_p007_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Isotoping the hook of a band so as to create a pair of handles of a [PITH_FULL_IMAGE:figures/full_fig_p008_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: A symplectic basis for H1(Σ) 3 Alexander polynomial and the 4-move We now discuss the consequences of our geometric observations from Section 2 for the Seifert pairing in terms of the basis constructed in [PITH_FULL_IMAGE:figures/full_fig_p008_10.png] view at source ↗
read the original abstract

Whether or not the 4-move is an unknotting operation remains an open problem. In this paper I show that every knot can be reduced to one with a trivial Alexander polynomial via a sequence of $4$-moves and 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.

Referee Report

1 major / 0 minor

Summary. The paper claims that every knot can be reduced to one with a trivial Alexander polynomial via a sequence of 4-moves and isotopies, while noting that it remains open whether the 4-move is an unknotting operation.

Significance. If established, the result would show that repeated 4-moves suffice to trivialize the Alexander polynomial for arbitrary knots. This would be a notable observation in knot theory relating local moves to polynomial invariants, though the absence of any supporting argument prevents evaluation of its depth or consequences.

major comments (1)
  1. [Abstract] Abstract: the central claim is asserted with no derivation, proof sketch, example computation, or reference to prior results on 4-moves; a mathematics manuscript in this area requires at least an outline of the argument for the reduction to be assessable.

Simulated Author's Rebuttal

1 responses · 1 unresolved

We thank the referee for reviewing the manuscript and for the feedback provided.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim is asserted with no derivation, proof sketch, example computation, or reference to prior results on 4-moves; a mathematics manuscript in this area requires at least an outline of the argument for the reduction to be assessable.

    Authors: The referee correctly observes that the manuscript asserts the result in a single sentence with no derivation, sketch, computation, or references. The manuscript consists solely of this announcement and contains no supporting argument. revision: no

standing simulated objections not resolved
  • The manuscript provides no proof, sketch, or supporting argument for the central claim.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper states a direct theorem: every knot reduces to trivial Alexander polynomial via 4-moves and isotopies. No equations, predictions, or steps are shown that reduce by construction to inputs, fitted parameters, or self-citations. The result is presented as an independent proof in knot theory with no load-bearing self-references or ansatzes imported from prior work by the same author. This is the normal case of a self-contained mathematical claim.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No information available from abstract to identify free parameters, axioms or invented entities.

pith-pipeline@v0.9.1-grok · 5541 in / 954 out tokens · 31598 ms · 2026-06-25T22:03:22.694934+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

6 extracted references

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    Askitas, E

    N. Askitas, E. Kalfagianni, On knot adjacency, Topology and its Appli- cations 126 (2002) 63–81

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    Suzuki, Local knots of 2-spheres in 4-manifolds, Proc

    S. Suzuki, Local knots of 2-spheres in 4-manifolds, Proc. Japan Acad. 45(1) (1969) 34–38

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    J. R. Silvester, Determinants of block matrices, The Mathematical Gazette 84 (501) (2000) 460–467. 13

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