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arxiv: 2605.04433 · v2 · pith:FO32WKG6new · submitted 2026-05-06 · 🧮 math.GT

Implementation of the Habegger--Lin decision algorithm

Yuka Kotorii , Atsuhiko Mizusawa This is my paper

Pith reviewed 2026-05-20 23:57 UTC · model grok-4.3

classification 🧮 math.GT
keywords link-homotopyHabegger-Lin algorithmMilnor mu-bar invariantsstring linksgroup actions4-component links5-component linkslink classification
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The pith

The Habegger-Lin algorithm, implemented for four- and five-component links, identifies link-homotopy classes that Milnor mu-bar invariants cannot distinguish.

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

Habegger and Lin classified link-homotopy classes of links by reducing the problem to classifying string links up to specific group actions, which yields a decision algorithm for whether two given links are link-homotopic. Prior explicit computation of those group actions for the four- and five-component cases makes the algorithm effective in those settings. This paper supplies a working implementation of the algorithm together with concrete pairs of links that are not link-homotopic yet share identical Milnor mu-bar invariants.

Core claim

Habegger and Lin gave a classification of link-homotopy classes of links in terms of that of string links modulo certain group actions. As an application they constructed an algorithm for determining whether given two links are link-homotopic. For the 4- and 5-component cases the group actions were computed explicitly, enabling the implementation which produces new pairs of links that are not link-homotopic but cannot be distinguished by Milnor's mu-bar invariants.

What carries the argument

The Habegger-Lin decision algorithm, which reduces the question of link-homotopy equivalence to the equivalence of string links under explicit group actions.

Load-bearing premise

The explicit computations of the group actions on string links for the four- and five-component cases are accurate and correctly translated into the implementation.

What would settle it

A geometric proof that any one of the reported pairs is actually link-homotopic would show that the implementation misclassifies that pair.

Figures

Figures reproduced from arXiv: 2605.04433 by Atsuhiko Mizusawa, Yuka Kotorii.

Figure 1
Figure 1. Figure 1: The closure of an n-component string link The closure map is surjective up to ambient isotopy. Hence, any link can be represented as the closure of a string link. Indeed, for any n-component link L, by cutting each component at a point, we obtain an n-component string link whose closure is L. However, this representation is not unique. For the link-homotopy case, Habegger and Lin [3] showed the Markov-type… view at source ↗
Figure 2
Figure 2. Figure 2: Local structures T1, T2 and Tn 1 2 3 4 y12 y13 y14 y23 y24 y34 y123 y124 y134 y124 y1234 y1324 view at source ↗
Figure 3
Figure 3. Figure 3: The canonical form for 4-component string links view at source ↗
Figure 4
Figure 4. Figure 4: A 4-component string link in the canonical form view at source ↗
Figure 5
Figure 5. Figure 5: 5-component string links L and L ′ Using our implementation, it is shown (in Step 4) that L and L ′ are not link-homotopic, although they cannot be distinguished by Milnor’s link-homotopy invariants. Indeed, µ(12) = 1 and the others are 0 or 0 mod 1 for both links. 5 view at source ↗
Figure 6
Figure 6. Figure 6: 5-component string links L and L ′ Using our implementation, it is shown (in Step 3) that L and L ′ are not link-homotopic, although they cannot be distinguished by Milnor’s link-homotopy invariants. Indeed, we can check that all Milnor invariants of length 2 are 0, µ(123) = µ(124) = 1, and the other Milnor invariants of length 3 are 0. Moreover, µ(1345) = µ(1435) = µ(2345) = µ(2435) = 0 and the other Miln… view at source ↗
read the original abstract

Habegger and Lin gave a classification of link-homotopy classes of links in terms of that of string links modulo certain group actions. As an application, they constructed an algorithm for determining whether given two links are link-homotopic. In \cite{KM4}, we explicitly computed these group actions for the 4- and 5-component cases. Consequently, the Habegger--Lin algorithm can be effectively applied in these cases. In this paper, we present an implementation of this algorithm, which is available at \cite{KMcode}, and exhibit new pairs of links that are not link-homotopic yet cannot be distinguished by Milnor's link-homotopy invariants, called $\overline{\mu}$-invariants.

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

0 major / 2 minor

Summary. The manuscript presents an implementation of the Habegger-Lin decision algorithm for classifying link-homotopy classes of 4- and 5-component links. It relies on the authors' prior explicit computation of the relevant group actions in the cited work KM4, makes the resulting code available at KMcode, and uses the implementation to exhibit new pairs of links that are not link-homotopic yet share identical Milnor mu-bar invariants.

Significance. If the implementation faithfully encodes the group actions from KM4, the work supplies a practical, reproducible computational tool for applying the Habegger-Lin procedure in the 4- and 5-component cases, where direct use of the original algorithm was previously intractable. The public release of the code at KMcode is a clear strength that supports independent verification and further experimentation. The new examples concretely illustrate that Milnor's mu-bar invariants do not separate all link-homotopy classes even for small numbers of components.

minor comments (2)
  1. The abstract states that new pairs are exhibited but does not indicate how many such pairs were found or give a brief description of one example; adding this information would help readers gauge the scope of the new phenomena without immediately consulting the code repository.
  2. The citation style for the code repository KMcode should be made consistent with the bibliographic entries for KM4 and other references.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive report, accurate summary of the manuscript's contributions, and the recommendation to accept. No major comments were raised.

Circularity Check

1 steps flagged

Implementation depends on prior self-computed group actions in KM4

specific steps
  1. self citation load bearing [Abstract]
    "In [KM4], we explicitly computed these group actions for the 4- and 5-component cases. Consequently, the Habegger--Lin algorithm can be effectively applied in these cases. In this paper, we present an implementation of this algorithm..."

    The decision procedure and new examples for 4- and 5-component links are made possible only by the authors' own prior explicit computation of the relevant group actions; the present work therefore inherits its concrete applicability from that self-cited result rather than re-deriving or independently verifying the actions.

full rationale

The paper presents a concrete implementation of the external Habegger-Lin algorithm together with new link examples. Its applicability to 4- and 5-component links rests on the explicit group-action computations published by the same authors in the cited prior work KM4. This constitutes a single load-bearing self-citation for the concrete cases treated, but the implementation itself, the code release, and the reported examples remain independently checkable and do not reduce the claimed results to a definitional tautology or fitted input. No other circular patterns appear in the derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on the existing Habegger-Lin classification and the authors' prior explicit group-action calculations without introducing new free parameters, axioms beyond standard group theory, or invented entities.

axioms (1)
  • domain assumption The Habegger-Lin classification of link-homotopy classes via string links and group actions holds for the cases under consideration.
    Invoked to justify that the algorithm correctly decides link-homotopy once the group actions are known.

pith-pipeline@v0.9.0 · 5642 in / 1257 out tokens · 73776 ms · 2026-05-20T23:57:05.848198+00:00 · methodology

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Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages

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