arxiv: 2605.03062 · v1 · submitted 2026-05-04 · 🧮 math.GT

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Gordian distance and clasper surgery for links

Anthony Bosman, Christopher W. Davis, Katherine Vance, Taylor Martin

Pith reviewed 2026-05-08 02:48 UTC · model grok-4.3

classification 🧮 math.GT

keywords C_k-equivalenceclasper surgeryGordian distanceMilnor invariantsunlinking numberlinking numberfinite type invariantscrossing changes

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The pith

Any n-component link with zero linking number can be reduced to a C_k-trivial link in at most n² crossing changes.

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

The paper shows that any n-component link whose pairwise linking numbers all vanish can always be turned into a C_k-trivial link by a sequence of at most n² crossing changes, and this upper bound holds for every natural number k. This extends the one-crossing-change result known for single knots and isolates the linking number as the only C_2-invariant that imposes a genuine lower bound on unlinking. Because Milnor invariants are preserved by C_k-equivalence for sufficiently large k, they therefore supply only partial information about the minimal number of crossing changes needed to unlink a link. The authors also construct explicit families of n-component links whose distance to C_k-triviality grows quadratically with n, showing the bound is asymptotically sharp. When linking numbers are nonzero and no component is already C_k-trivial, the exact crossing-change distance is computed.

Core claim

We prove that any n-component link L with all pairwise linking numbers zero satisfies that the minimal number of crossing changes needed to reach a C_k-trivial link is at most n² for every k. We exhibit a sequence of n-component links realizing a quadratic lower bound on this distance. As a direct consequence, Milnor invariants, which are invariants of C_k-equivalence, carry only limited information about the unlinking number. When the linking numbers are nonzero and no component is C_k-trivial, we determine the precise number of crossing changes required to reach a C_k-trivial link.

What carries the argument

The Gordian distance (minimal crossing changes) from a link to a C_k-trivial link, where C_k-equivalence is the geometric relation generated by degree-k clasper surgeries.

If this is right

Milnor invariants give only limited information about the unlinking number of a link.
The n² upper bound on distance to C_k-triviality is independent of k.
There exist n-component links requiring quadratically many crossing changes to reach C_k-triviality.
When linking numbers are nonzero and no component is C_k-trivial, the exact crossing-change distance is determined.

Where Pith is reading between the lines

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

The quadratic control on distance to C_k-triviality may allow similar bounds for other finite-type invariants of links.
One could check whether the same n² ceiling holds when the target is replaced by other filtrations such as concordance or other surgery equivalences.
The construction of quadratic lower-bound examples may generalize to produce families that remain far from C_k-triviality under additional geometric constraints such as fixed volume or bounded crossing number.

Load-bearing premise

The constructions assume that links are standardly embedded in the 3-sphere and that clasper surgery and crossing-change operations interact exactly as described in the existing literature on C_k-equivalence.

What would settle it

An explicit n-component link with all pairwise linking numbers zero whose minimal number of crossing changes to any C_k-trivial link exceeds n² for some fixed k.

Figures

Figures reproduced from arXiv: 2605.03062 by Anthony Bosman, Christopher W. Davis, Katherine Vance, Taylor Martin.

**Figure 2.** Figure 2: (a) A basic clasper, C, in the complement of a link J. (b) A 2- component link sitting in a neighborhood of C which inherits the blackboard framing. (c) Modifying J by surgery along C. 1.3. Outline of the paper. In Section 3, we recall the theory of claspers and Ck equivalence. We close the section with a summary of a few of the moves used in [9] to produce a group structure on Ck-trivial links up to Ck+1 … view at source ↗

**Figure 3.** Figure 3: (a) A clasper c showing its decomposition into bands and constituents. The middle band has a right handed half twist. The set of constituents includes one box, one node, and four leaves. (b) The same clasper, encoded as a union of leaves, nodes, boxes, and framed arcs between them. The “+” indicates a positive half twist difference from the blackboard framing. A “−” will be used for a negative half twist… view at source ↗

**Figure 4.** Figure 4: Transforming a clasper into a collection of simple claspers. from the blackboard framing by a positive (or negative) half twist we will decorate it with a + sign (or − sign). In order to perform surgery along a clasper we first replace each disk leaf with a collar neighborhood of its boundary, transforming it to an annular leaf, and then perform the move of Figure 4a at each box and the move of Figure 4b a… view at source ↗

**Figure 5.** Figure 5: Two moves which preserve the result of clasper surgery. The degree of a tree-shaped clasper is defined to be the number of nodes in that clasper plus 1 (or equivalently, the number of leaves minus 1.) The degree of a forest is the minimal degree among all of its trees. If L is a link, c is a degree k simple forest for L, and L c is the result of changing L by surgery along c, then we say that L and L c are… view at source ↗

**Figure 9.** Figure 9: Left: A 2-component link L with µ1122(L) = 6 and whose every component is a trefoil. The “+6” in the box indicates 6 full right handed twists and the T indicates tying in a trefoil knot. Right: a 3-component link whose every 2-component link is the link of (a) and with µ123(L) = ±1 Proof. Consider any sequence of crossing changes transforming L to a C3-trivial link. Since each component is not C3-trivial, … view at source ↗

read the original abstract

In 2000, Habiro introduced the notion of $C_k$-equivalence of knots and links. This geometric filtration is closely connected to finite type invariants, a class of invariants including Milnor's invariants. Shortly thereafter, Ohyama, Taniyama, and Yamada proved that $C_k$-equivalence, and by extension finite type invariants, say very little about the unknotting number by showing that any knot is at most one crossing change away from being $C_k$-trivial for any $k\in \mathbb{N}$. The same is not true for links, since the pairwise linking number gives a lower bound on unlinking and is an invariant of $C_2$-equivalence. We prove that, aside from the linking number, the result of Ohyama, Taniyama, and Yamada extends to links: any $n$-component link with linking number zero can be reduced to a $C_k$-trivial link in at most $n^2$ crossing changes. As a consequence, Milnor's invariants carry only limited information about the unlinking number. To establish a lower bound, we produce a sequence of $n$-component links for which the crossing change distance to a $C_k$-trivial link grows quadratically in $n$. Notably, these bounds are independent of the choice of $k\in \mathbb{N}$. Finally, we determine the exact number of crossing changes to a $C_k$-trivial link for links with nonzero linking numbers and where no component is $C_k$-trivial.

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

The paper extends the Ohyama-Taniyama-Yamada knot result to links by proving n² crossing changes suffice for zero-linking-number n-component links to reach C_k-triviality for any k, with a quadratic lower bound on examples and an exact count when linking is nonzero.

read the letter

The key result is that n-component links with zero linking number reach C_k-triviality in at most n squared crossing changes, with a matching quadratic lower bound on a family of examples, and this holds for every k. When the linking number is nonzero, they give the exact count. This is a direct and useful extension of the Ohyama-Taniyama-Yamada knot theorem to the link setting. The independence from k is particularly clean, and the consequence that Milnor invariants have limited bearing on unlinking numbers follows immediately. The constructions appear to use standard crossing changes and clasper operations in the 3-sphere, building on Habiro's work without introducing new machinery. The paper does well in giving both upper and lower bounds explicitly and in separating the zero and nonzero linking cases. The citation pattern is appropriate, pointing back to the foundational papers on C_k-equivalence. The soft spots are minor. The lower bound is shown only for one sequence of links rather than characterizing the distance for all links, but that is common in these distance problems. The assumptions about embeddings and move behaviors are the usual ones in the field. This work is aimed at researchers in low-dimensional topology who care about finite-type invariants and geometric distances between links. Someone studying unlinking numbers or Milnor invariants would find the explicit bounds helpful. It is solid enough to deserve a serious referee. I would send it to peer review.

Referee Report

0 major / 4 minor

Summary. The paper extends the Ohyama–Taniyama–Yamada theorem from knots to links by proving that any n-component link in S³ with vanishing pairwise linking numbers can be reduced to a C_k-trivial link by at most n² crossing changes, for arbitrary k. It supplies a quadratic lower bound via an explicit family of examples, determines the exact crossing-change distance to C_k-triviality when linking numbers are nonzero (and no component is already C_k-trivial), and concludes that Milnor invariants beyond the linking numbers give only limited information about the unlinking number. The arguments rely on standard properties of clasper surgery and crossing changes.

Significance. If the stated bounds hold, the result is significant: it shows that C_k-equivalence (and hence finite-type invariants) is strictly coarser than the Gordian distance for links once linking numbers vanish, with the gap growing quadratically in the number of components. The k-independence of the bounds and the matching lower-bound construction are notable strengths, as is the exact determination for the nonzero-linking case. The work cleanly separates the linking-number obstruction from higher-order Milnor invariants.

minor comments (4)

§1, paragraph after Definition 1.2: the phrase 'Gordian distance to C_k-triviality' is introduced without an explicit reference to the crossing-change metric; a one-sentence reminder of the definition would improve readability for readers outside the immediate area.
Theorem 1.3 (nonzero-linking case): the statement assumes 'no component is C_k-trivial,' but the proof sketch in §4 does not explicitly verify that this hypothesis is preserved under the sequence of moves; adding a short sentence confirming invariance would remove any ambiguity.
Figure 3 (the quadratic lower-bound family): the caption does not indicate the value of k for which the diagram is drawn; since the bound is claimed independent of k, a parenthetical note '(for any k ≥ 2)' would clarify the figure's role.
References: the citation to Habiro's original C_k paper is listed as [Hab00], but the bibliography entry lacks the journal volume and page range; this is a minor formatting omission.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation of minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's central theorems extend the externally cited Ohyama-Taniyama-Yamada result on knots to links via standard geometric properties of clasper surgery and crossing changes in S^3, as established in Habiro's prior work. The n² upper bound, quadratic lower bound examples, and exact counts for nonzero linking number are derived from explicit constructions and reductions that do not reduce to self-definitions, fitted inputs, or load-bearing self-citations. All load-bearing facts about C_k-equivalence and finite-type invariants are imported from independent literature without internal circular dependence on the present claims.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the claims rest on standard definitions and operations from geometric topology; no free parameters, new entities, or ad-hoc axioms are introduced.

axioms (1)

domain assumption Standard properties of links in the 3-sphere and the behavior of clasper surgery and crossing changes under C_k-equivalence
The paper invokes Habiro's C_k-equivalence and the earlier results of Ohyama-Taniyama-Yamada without re-deriving them.

pith-pipeline@v0.9.0 · 5582 in / 1410 out tokens · 75700 ms · 2026-05-08T02:48:11.005491+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

23 extracted references · 1 canonical work pages

[1]

On the indeterminacy of Milnor’s triple linking number.J

Jonah Amundsen, Eric Anderson, and Christopher William Davis. On the indeterminacy of Milnor’s triple linking number.J. Knot Theory Ramifications, 29(9):15, 2020. Id/No 2050064

2020
[2]

The C-complex clasp number of links.Rocky Mt

Jonah Amundsen, Eric Anderson, Christopher William Davis, and Daniel Guyer. The C-complex clasp number of links.Rocky Mt. J. Math., 50(3):839–850, 2020

2020
[3]

How many crossing changes or delta-moves does it take to get to a homotopy trivial link? 2025

Anthony Bosman, Christopher William Davis, Taylor Martin, Carolyn Otto, and Katherine Vance. How many crossing changes or delta-moves does it take to get to a homotopy trivial link? 2025. ArXiv: https://arxiv.org/abs/2412.18075. To appear inAlgebraic and Geometric Topology

work page arXiv 2025
[4]

Whitney tower concordance of classical links

James Conant, Rob Schneiderman, and Peter Teichner. Whitney tower concordance of classical links. Geom. Topol., 16(3):1419–1479, 2012

2012
[5]

Clasper concordance, whitney towers and re- peating milnor invariants.J

James Conant, Rob Schneiderman, and Peter Teichner. Clasper concordance, whitney towers and re- peating milnor invariants.J. Knot Theory Ramifications, 34(4):2550013, 2025

2025
[6]

Davis, Matthias Nagel, Patrick Orson, and Mark Powell

Christopher W. Davis, Matthias Nagel, Patrick Orson, and Mark Powell. Surface systems and triple linking numbers.Indiana Univ. Math. J., 69(7):2505–2547, 2020

2020
[7]

The classification of links up to link-homotopy.J

Nathan Habegger and Xiao-Song Lin. The classification of links up to link-homotopy.J. Amer. Math. Soc., 3(2):389–419, 1990

1990
[8]

On link concordance and Milnor’s µinvariants.Bull

Nathan Habegger and Xiao-Song Lin. On link concordance and Milnor’s µinvariants.Bull. London Math. Soc., 30(4):419–428, 1998

1998
[9]

Claspers and finite type invariants of links.Geom

Kazuo Habiro. Claspers and finite type invariants of links.Geom. Topol., 4:1–83 (electronic), 2000

2000
[10]

ProQuest LLC, Ann Arbor, MI, 1988

Gyo Taek Jin.Invariants of two-component links. ProQuest LLC, Ann Arbor, MI, 1988. Thesis (Ph.D.)– Brandeis University

1988
[11]

Two-bridge knots with unknotting number one.Proc

Taizo Kanenobu and Hitoshi Murakami. Two-bridge knots with unknotting number one.Proc. Amer. Math. Soc., 98(3):499–502, 1986

1986
[12]

Unlinking, splitting, and some other NP-hard problems in knot theory

Dale Koenig and Anastasiia Tsvietkova. Unlinking, splitting, and some other NP-hard problems in knot theory. InProceedings of the 32nd annual ACM-SIAM symposium on discrete algorithms, SODA 2021, Alexandria, VA, USA, virtual, January 10–13, 2021, pages 1496–1507. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM); New York, NY: Associ...

2021
[13]

Two-bridge links with unlinking number one.Proc

Peter Kohn. Two-bridge links with unlinking number one.Proc. Amer. Math. Soc., 113(4):1135–1147, 1991

1991
[14]

Unlinking two component links.Osaka J

Peter Kohn. Unlinking two component links.Osaka J. Math., 30(4):741–752, 1993

1993
[15]

Link-homotopy classes of 4-component links, claspers and the Habegger-Lin algorithm.J

Yuka Kotorii and Atsuhiko Mizusawa. Link-homotopy classes of 4-component links, claspers and the Habegger-Lin algorithm.J. Knot Theory Ramifications, 32(6):Paper No. 2350045, 26, 2023

2023
[16]

W. B. Raymond Lickorish. The unknotting number of a classical knot. InCombinatorial methods in topology and algebraic geometry (Rochester, N.Y., 1982), volume 44 ofContemp. Math., pages 117–121. Amer. Math. Soc., Providence, RI, 1985

1982
[17]

Enhanced linking numbers.The American Mathematical Monthly, 110(5):361–385, 2003

Charles Livingston. Enhanced linking numbers.The American Mathematical Monthly, 110(5):361–385, 2003

2003
[18]

Link groups.Ann

John Milnor. Link groups.Ann. of Math. (2), 59:177–195, 1954

1954
[19]

Isotopy of links

John Milnor. Isotopy of links. Algebraic geometry and topology. InA symposium in honor of S. Lefschetz, pages 280–306. Princeton University Press, Princeton, N. J., 1957

1957
[20]

On a certain move generating link-homology.Math

Hitoshi Murakami and Yasutaka Nakanishi. On a certain move generating link-homology.Math. Ann., 284(1):75–89, 1989. GORDIAN DISTANCE AND CLASPER SURGERY FOR LINKS 13

1989
[21]

A note on unknotting number.Math

Yasutaka Nakanishi. A note on unknotting number.Math. Sem. Notes Kobe Univ., 9(1):99–108, 1981

1981
[22]

Web diagrams and realization of Vassiliev invariants by knots.J

Yoshiyuki Ohyama. Web diagrams and realization of Vassiliev invariants by knots.J. Knot Theory Ramifications, 9(5):693–701, 2000

2000
[23]

Realization of Vassiliev invariants by unknot- ting number one knots.Tokyo J

Yoshiyuki Ohyama, Kouki Taniyama, and Shuji Yamada. Realization of Vassiliev invariants by unknot- ting number one knots.Tokyo J. Math., 25(1):17–31, 2002. Department of Mathematics and Physics, Andrews University Email address:bosman@andrews.edu URL:andrews.edu/cas/math/faculty/bosman-anthony.html Department of Mathematics, University of Wisconsin–Eau Cl...

2002