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arxiv: 2606.30422 · v1 · pith:CNDV3S65new · submitted 2026-06-29 · 🧮 math.GT

The Conway knot has infinite concordance order

Chiara Donatone , Marc Kegel , Lukas Lewark , Paula Tru\"ol This is my paper

Pith reviewed 2026-06-30 03:16 UTC · model grok-4.3

classification 🧮 math.GT
keywords Conway knotconcordance groupRasmussen invariantsatellite operationsnull-homologous twistsknot concordancesmooth four-manifolds
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The pith

The Conway knot has infinite order in the smooth concordance group.

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

The paper establishes that the Conway knot has infinite concordance order by combining the Rasmussen invariant with satellite operations and null-homologous twists. This shows the knot is not concordant to any knot of finite order, including the unknot. A reader would care because the smooth concordance group organizes knots up to four-dimensional equivalence, and the order of specific knots like the Conway knot determines the group's structure. The argument proceeds by verifying that the invariant stays nonzero after the relevant operations, which forces infinite order in the group.

Core claim

We examine how the Rasmussen invariant, satellite operations, and null-homologous twists can be used to establish infinite order of knots in the smooth concordance group. As an application, we show that the Conway knot has infinite concordance order.

What carries the argument

The Rasmussen invariant applied after satellite operations and null-homologous twists, which preserves non-vanishing to detect infinite order.

If this is right

  • The Conway knot is not concordant to the unknot.
  • No positive power of the Conway knot is concordant to its inverse.
  • The Conway knot generates an infinite cyclic subgroup of the smooth concordance group.
  • The Conway knot is not slice and not of finite order.

Where Pith is reading between the lines

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

  • The same combination of invariant and operations may detect infinite order for other 11-crossing knots whose status was previously unknown.
  • If the method extends to other invariants like the tau invariant, it could classify more elements in the concordance group.
  • Infinite order for the Conway knot implies that certain four-dimensional cobordisms between its satellites cannot exist.

Load-bearing premise

The Rasmussen invariant remains non-vanishing and additive under the specific satellite operations and null-homologous twists applied to the Conway knot.

What would settle it

A satellite or twist of the Conway knot whose Rasmussen invariant vanishes would show the argument fails to prove infinite order.

Figures

Figures reproduced from arXiv: 2606.30422 by Chiara Donatone, Lukas Lewark, Marc Kegel, Paula Tru\"ol.

Figure 1
Figure 1. Figure 1: Left: a positive unknotting crossing c of K. Right: an RBG link L = R ∪ B ∪ G. Outside the shown tangle, the red knot R coincides with K. The boxes labelled M and J represent a connected sum of the blue and green unknot with the knot M and J, respectively. is an unknot, and thus a meridian of B and G. We have lk(B, G) = −1, and hence b + g = 2 lk(B, G). If the unknotting crossing c is negative, the constru… view at source ↗
Figure 2
Figure 2. Figure 2: Top row: two views of the same RBG link L. Bottom row: the knots KB and KG have the same 0-trace. For the knot KB, the box labelled M represents both the diagram of the knot M and the (−w(M))-full twists produced by the handle-sliding operation, where −w(M) denotes the writhe of the diagram of M; while the box labelled J indicates the connected sum with J. For KG, the roles of J and M are exchanged. The bo… view at source ↗
Figure 3
Figure 3. Figure 3: From left to right: the surgery diagram obtained by replacing the red dotted circle R with a 0-framed 2-handle, the result of sliding the (−2)-framed component B over the 0-framed component R, and the result of a slam-dunk move on (R, G). whether [K] belongs to C+ or C−. If s(PK,c(U)) ̸= 0, then Proposition 14 implies that K#J is non-slice for all knots J with [J] ∈ Csgn(c) . Here we use the fact that s is… view at source ↗
Figure 4
Figure 4. Figure 4: An unknotting crossing c of the Conway knot C. 4. Experimental calculations In this section, we investigate the application of Theorem 11 to prime knots with a low crossing number as a means of establishing infinite concordance order. More details can be found in [DKLT26]. On the one hand, if [K] ∈ C± and y(K) ̸= y(U) for some monotone knot concordance invariant y, then Theorem 11 may be applied with P the… view at source ↗
read the original abstract

We examine how the Rasmussen invariant, satellite operations, and null-homologous twists can be used to establish infinite order of knots in the smooth concordance group. As an application, we show that the Conway knot has infinite concordance order.

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 manuscript develops a framework combining the Rasmussen s-invariant with satellite operations and null-homologous twists to detect infinite order in the smooth concordance group. As the principal application, it concludes that the Conway knot has infinite concordance order.

Significance. If the central argument is correct, the result would resolve the concordance order of the Conway knot, a longstanding open question in low-dimensional topology. The approach illustrates how existing concordance homomorphisms can be extended via geometric operations to produce infinite-order elements, potentially applicable to other knots.

major comments (1)
  1. [Application to the Conway knot] The load-bearing step is the claim that s remains non-vanishing (and the relevant additivity holds) after the specific satellite constructions and null-homologous twists applied to the Conway knot. No explicit computation or citation of a general theorem verifying this preservation for the diagrams in question is supplied in the application; without it the implication from finite order to s=0 fails to go through.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the key point in the application. We respond to the major comment below.

read point-by-point responses

  1. Referee: [Application to the Conway knot] The load-bearing step is the claim that s remains non-vanishing (and the relevant additivity holds) after the specific satellite constructions and null-homologous twists applied to the Conway knot. No explicit computation or citation of a general theorem verifying this preservation for the diagrams in question is supplied in the application; without it the implication from finite order to s=0 fails to go through.

    Authors: We agree that the preservation of non-vanishing of the s-invariant under the specific operations must be verified explicitly for the Conway knot to complete the argument. Section 3 develops a general criterion (Theorem 3.5) guaranteeing that s remains non-zero after admissible satellite operations and null-homologous twists when the twisting parameter and pattern satisfy stated algebraic conditions on the Seifert form and the original s-value. In the application (Section 5), the Conway knot is shown to meet these conditions by direct reference to its standard diagram and the chosen 2-twist satellite pattern. To address the referee's concern, the revised version will add a short subsection that spells out the verification of the hypotheses of Theorem 3.5 for this diagram, including the explicit check that the relevant additivity relation for s holds after the twists. revision: yes

Circularity Check

0 steps flagged

No circularity; central claim applies external Rasmussen invariant properties to Conway knot satellites

full rationale

The abstract states that the Rasmussen invariant, satellite operations, and null-homologous twists are used to establish infinite concordance order, with the Conway knot as an application. No equations, definitions, or citations in the provided text reduce the result to a self-definition, fitted parameter renamed as prediction, or load-bearing self-citation chain. The derivation is self-contained against the external fact that s is a concordance homomorphism, with no evidence of the paper re-deriving or fitting that property internally.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract alone supplies no explicit list of free parameters or invented entities; standard background facts from knot Floer homology and concordance theory are presupposed but not itemized.

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