arxiv: 2605.04433 · v1 · submitted 2026-05-06 · 🧮 math.GT

Recognition: unknown

Implementation of the Habegger--Lin decision algorithm

Atsuhiko Mizusawa, Yuka Kotorii

Pith reviewed 2026-05-08 17:26 UTC · model grok-4.3

classification 🧮 math.GT MSC 57M2557K10

keywords link-homotopyHabegger-Lin algorithmstring linksMilnor invariantslink classificationcomputational topologyfour-component linksfive-component links

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The pith

Implementation of the Habegger-Lin algorithm decides link-homotopy for four- and five-component links and finds new examples beyond Milnor invariants.

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

Habegger and Lin reduced the classification of link-homotopy classes of links to the classification of string links modulo certain group actions and gave a decision algorithm based on that reduction. The authors had already computed the explicit form of those group actions for the four- and five-component cases. This paper supplies a working implementation of the resulting procedure and applies it to produce concrete pairs of links that share all Milnor μ-bar invariants yet are not link-homotopic. A reader cares because the work turns an abstract existence proof into a usable test and exhibits the first computable counterexamples to the completeness of the standard invariants.

Core claim

The Habegger-Lin decision procedure, which determines whether two links are link-homotopic by testing equivalence of associated string links under the computed group actions, has been implemented for four- and five-component links. When run on sample inputs, the procedure yields new pairs of links that are not link-homotopic but cannot be separated by any of Milnor's μ-bar invariants.

What carries the argument

The Habegger-Lin decision algorithm that reduces link-homotopy of links to equivalence of string links modulo explicit group actions.

If this is right

The implementation supplies a practical test for link-homotopy of any given four- or five-component links.
Milnor's μ-bar invariants fail to classify link-homotopy classes completely once the number of components reaches four or five.
The code base makes it feasible to search systematically for further examples where the invariants are insufficient.
Any two links declared non-homotopic by the program are genuinely non-homotopic provided the prior group-action tables are accurate.

Where Pith is reading between the lines

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

Once analogous group actions are computed for six or more components, the same implementation strategy would decide link-homotopy in those cases as well.
The concrete pairs found here could be used to test whether other proposed invariants or geometric constructions detect the same distinctions.
Public availability of the code allows independent verification and extension by researchers studying low-dimensional link problems.

Load-bearing premise

The explicit computations of the group actions on string links for the four- and five-component cases reported in the prior work are correct and complete.

What would settle it

An independent check, by hand or by another program, showing that one of the exhibited pairs is actually link-homotopic or that the implementation returns the wrong answer on a pair whose status is already known.

Figures

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

**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: 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: The canonical form for 4-component string links view at source ↗

**Figure 4.** Figure 4: A 4-component string link in the canonical form view at source ↗

**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: 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.

Desk Editor's Note private letter to a colleague

This paper ships runnable code for the Habegger-Lin decision procedure on 4- and 5-component links and produces new explicit pairs that Milnor invariants miss, but the whole thing sits on top of the authors' own earlier group-action tables with no fresh checks.

read the letter

The paper implements the Habegger-Lin algorithm for deciding link-homotopy of links with four or five components and exhibits concrete pairs that are not link-homotopic yet share the same Milnor mu-bar invariants. The code is made available at the cited repository, which turns the abstract procedure into something people can actually run on examples. That is the main advance: a working tool plus new instances that demonstrate the invariants are incomplete in these cases. The underlying classification comes from Habegger-Lin, and the group actions on string links come from the authors' prior computation, so the new material is the implementation step and the output pairs. The work is narrow but fills a practical gap for anyone who needs to classify small-component links up to homotopy. It does not reorganize the theory or handle larger numbers of components. The soft spot is the complete dependence on the earlier group-action tables. This manuscript gives no test cases, no comparison against known results, no sample runs, and no independent verification that the actions were computed correctly. If those tables contain an omission, both the decision outputs and the claimed new pairs become unreliable. The abstract asserts that the implementation works and that the pairs exist, but supplies none of the supporting data that would let a reader check. This is for specialists already working on link-homotopy or computational 3-manifold topology who want to apply the Habegger-Lin method to specific 4- or 5-component examples. A reader outside that niche will not find new conceptual tools. It deserves peer review because the contribution is concrete and the code is public; referees can examine the implementation details and press on the dependence on the prior calculation.

Referee Report

2 major / 2 minor

Summary. The paper presents an implementation of the Habegger-Lin decision algorithm for determining whether two links are link-homotopic, based on the classification of string links modulo certain group actions. Building on the authors' explicit computations of these group actions for the 4- and 5-component cases in their prior work [KM4], the implementation is made publicly available at [KMcode], and the paper exhibits new pairs of links that are claimed to be link-homotopy inequivalent yet indistinguishable by all Milnor μ-bar invariants.

Significance. If the implementation is correct, the work is significant because it makes the Habegger-Lin procedure effective and computable for links with 4 or 5 components and supplies concrete examples showing that Milnor's μ-bar invariants do not classify link-homotopy classes in these cases. The public code availability is a positive feature that enables independent checks and further applications in geometric topology.

major comments (2)

[Abstract and implementation description] The abstract asserts that the implementation works and that new pairs exist, but the manuscript supplies no verification data, no description of test cases, no error analysis, and no comparison with independent methods. This is load-bearing for the central claim that the exhibited pairs are not link-homotopic.
[Introduction and reliance on [KM4]] The central result depends on the authors' own prior computation of the group actions on string links for the 4- and 5-component cases, reported in [KM4]. The overall claim inherits whatever verification status that earlier calculation possesses, with no independent derivation or cross-check against known low-complexity cases appearing here.

minor comments (2)

[References and code availability] The citation for the code repository [KMcode] would benefit from a specific version, commit hash, or stable URL to ensure long-term reproducibility.
[Introduction] A brief recap of the Habegger-Lin framework and the role of the group actions would improve accessibility for readers unfamiliar with the prior work [KM4].

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address each major comment point by point below and indicate the revisions we plan to make.

read point-by-point responses

Referee: [Abstract and implementation description] The abstract asserts that the implementation works and that new pairs exist, but the manuscript supplies no verification data, no description of test cases, no error analysis, and no comparison with independent methods. This is load-bearing for the central claim that the exhibited pairs are not link-homotopic.

Authors: We agree that additional verification details would strengthen the presentation of the central claims. The implementation is deterministic, being a direct realization of the Habegger--Lin procedure once the group actions are known, and the code is publicly available at [KMcode] for independent inspection. In the revised manuscript we will add a dedicated verification section that describes test cases on links with previously known link-homotopy status, supplies sample output for the new pairs, and notes that the algebraic nature of the computations eliminates floating-point error concerns. We will also record explicit comparisons with the Milnor invariants on these examples. revision: yes
Referee: [Introduction and reliance on [KM4]] The central result depends on the authors' own prior computation of the group actions on string links for the 4- and 5-component cases, reported in [KM4]. The overall claim inherits whatever verification status that earlier calculation possesses, with no independent derivation or cross-check against known low-complexity cases appearing here.

Authors: The present work is an implementation paper that applies the explicit group-action computations already obtained in [KM4]; a full re-derivation would lie outside its scope. To address the referee's concern we will insert a short subsection of cross-checks that applies the implementation to low-complexity cases (including 3-component links, where the classification is known to coincide with the Milnor invariants) and confirms that the output matches established theoretical expectations. This provides an independent sanity check on the code without duplicating the prior group-action calculations. revision: partial

Circularity Check

1 steps flagged

Implementation and new link examples depend on unverified self-computed group actions from prior paper [KM4]

specific steps

self citation load bearing [Abstract]
"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."

The implementation and exhibition of new pairs are presented as a direct consequence of the group actions computed in the authors' prior self-cited paper [KM4]. The current work provides no independent derivation or verification of those actions, so the validity of the decision procedure and the new examples reduces to the correctness of the self-citation.

full rationale

The paper's central contribution is an implementation of the Habegger-Lin algorithm together with concrete 4- and 5-component examples. This rests directly on the explicit group-action computations reported in the authors' own earlier work [KM4], which is cited as the enabling step without re-derivation, machine verification, or external cross-check in the present manuscript. The abstract explicitly states that the algorithm 'can be effectively applied' as a consequence of those prior computations. This creates a self-citation load-bearing dependency: if the action tables or orbit representatives in [KM4] contain an error, both the implementation's outputs and the claimed new link pairs become unreliable. No other circular patterns (self-definitional, fitted predictions, ansatz smuggling, etc.) are present.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The implementation rests on the Habegger-Lin classification theorem and on the authors' earlier explicit calculation of the relevant group actions; no new free parameters or invented entities are introduced in the abstract.

axioms (1)

domain assumption Habegger-Lin classification of link-homotopy classes via string links modulo group actions
The decision algorithm is derived directly from this theorem.

pith-pipeline@v0.9.0 · 5411 in / 1163 out tokens · 26813 ms · 2026-05-08T17:26:30.309736+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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