arxiv: 2603.08597 · v2 · submitted 2026-03-09 · 🧮 math.GT

Recognition: 1 theorem link

· Lean Theorem

The n-adjacency graph for knots

Marion Campisi , Brandy Doleshal , Eric Staron

Authors on Pith no claims yet

Pith reviewed 2026-05-15 13:29 UTC · model grok-4.3

classification 🧮 math.GT

keywords knot theoryn-adjacencycrossing changesgraph on knotsgeneralized crossing changes

0 comments

The pith

Knots form the n-adjacency graph Γ_n where edges exist when n crossing circles allow any nonempty subset change to reach the same target knot.

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

The paper defines a knot K as n-adjacent to knot K' if K contains a set of n crossing circles such that changing crossings in any nonempty subset of the set produces K'. It introduces the graph Γ_n with all knots as vertices and an edge between two knots precisely when they satisfy this n-adjacency relation. Several structural results about Γ_n are proved for different values of n. The construction supplies a uniform way to record how knots are related by coordinated groups of crossing changes rather than single changes.

Core claim

We define the n-adjacency graph Γ_n on the set of knots, with an edge from K to K' whenever K is n-adjacent to K' via some collection of n crossing circles, and we establish several results about the properties of these graphs.

What carries the argument

n-adjacency relation: a knot K is n-adjacent to K' when there exist n crossing circles in K such that a generalized crossing change on any nonempty subset of those circles produces exactly K'.

If this is right

The family of graphs Γ_n extends single-crossing adjacency to collections of n coordinated changes.
Results proved for Γ_n apply uniformly across all integers n and all knots.
The graph organizes knots according to the existence of multi-crossing operations that preserve the target type.

Where Pith is reading between the lines

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

The connected components of Γ_n may group knots that share similar unknotting or surgery properties.
Enumeration of edges in Γ_n for small knots could produce new tables of multi-change relations.
The construction invites comparison with other knot graphs such as the Gordian graph.

Load-bearing premise

The n-adjacency relation is well-defined and yields a consistent undirected graph structure under the standard rules for generalized crossing changes.

What would settle it

A concrete pair of knots together with a claimed set of n crossing circles for which changing one particular nonempty subset produces a knot different from the claimed target K'.

Figures

Figures reproduced from arXiv: 2603.08597 by Brandy Doleshal, Eric Staron, Marion Campisi.

**Figure 1.** Figure 1: K′ We take the crossing circles C1 and C2 to each enclose one of the crossings in the box containing m and n crossings, respectively. Insisting that β have odd 2-bridge length allows us to guarantee that the algebraic intersection number of Kβ(m, n) with the disk bounded by Ci is zero. In terms of the braid word for Kβ(m, n), we can think of doing the crossing change on C1 as deleting σ m 2 from the braid … view at source ↗

read the original abstract

A knot $K$ is called $n$-adjacent to a knot $K'$ if there is a set of $n$ crossing circles $\mathcal C$ in $K$ so that a generalized crossing change at any nonempty subset of crossings in $\mathcal C$ yields $K'$. In this paper, the authors define a new graph $\Gamma_n$ to represent $n$-adjacency relationships between knots. We prove several results about this new object.

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 defines the n-adjacency graph Γ_n on knots and proves some results about it, but the abstract gives almost no detail on what those results actually say.

read the letter

The main contribution is a new graph Γ_n whose vertices are knots and whose edges record when one knot is n-adjacent to another through a fixed set of n crossing circles. A generalized crossing change on any nonempty subset of those circles turns one knot into the other. They package the existing notion of n-adjacency into this graph and then claim several theorems about it. That construction itself is the fresh piece; earlier papers studied crossing changes and adjacency relations, but this one turns the relation into an explicit graph object with its own properties to investigate. The definition is clean and follows directly from the standard crossing-circle setup, so there is no obvious circularity or invented machinery. The paper does a straightforward job of stating the definition and moving to results without extra fluff. The soft spots are mostly about visibility. The abstract is so short that it never names even one of the proved statements, which makes it hard to judge whether the theorems are routine consequences of the definition or actually add something. From the given summary the relation is well-defined by construction, and the stress-test note found no internal contradiction, but without the proofs it is impossible to check for gaps or whether the claims are as strong as stated. This is aimed at knot theorists who already work with crossing changes and want a graph-theoretic language for them. Someone outside that niche will not get much from it. A reader who cares about combinatorial models in low-dimensional topology could extract value from the new object. I would send it to peer review because it introduces a concrete new construction with claimed theorems in an established area; the referee can sort out the depth of the results once the full proofs are in front of them.

Referee Report

0 major / 2 minor

Summary. The paper defines n-adjacency between knots K and K' via the existence of n crossing circles in K such that a generalized crossing change on any nonempty subset yields K'. It constructs the graph Γ_n with knots as vertices and edges for n-adjacency pairs, then proves several results on the graph's properties.

Significance. If the claimed results hold, Γ_n supplies a new graph-theoretic framework for studying multi-crossing-change relations among knots. This could connect to existing invariants such as unknotting number and crossing distance, offering a combinatorial lens on knot transformations that is grounded in standard crossing-circle techniques.

minor comments (2)

The abstract states that several results are proved but does not name even one theorem; adding a brief statement of the main result would improve readability.
In the definition of n-adjacency, confirm whether the relation is symmetric (i.e., whether K n-adjacent to K' implies the converse) so that it is clear whether Γ_n is undirected.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their concise summary of our definition of n-adjacency and the graph Γ_n, as well as for the positive assessment of its potential connections to unknotting number and crossing distance. We appreciate the recommendation of minor revision. Since the report contains no specific major comments requiring changes, we have prepared the following responses.

Circularity Check

0 steps flagged

No significant circularity detected in the n-adjacency graph definition

full rationale

The paper defines n-adjacency via a standard construction using crossing circles and generalized crossing changes, then introduces the graph Γ_n on knots based on this relation and proves results about the resulting object. No derivation step reduces by construction to a fitted parameter, self-citation chain, or renaming of prior results; the central claims rest on the new definition together with ordinary knot-theoretic operations that are independent of the paper's own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The work rests on standard knot theory definitions of crossings and crossing changes; no free parameters or invented entities beyond the new graph itself are apparent from the abstract.

axioms (1)

standard math Standard axioms and definitions of knot theory, including crossing changes and knot equivalence
The n-adjacency definition directly invokes generalized crossing changes, which are part of established knot theory.

invented entities (1)

n-adjacency graph Γ_n no independent evidence
purpose: To represent n-adjacency relationships between knots
Newly introduced object whose properties are studied in the paper.

pith-pipeline@v0.9.0 · 5362 in / 1099 out tokens · 36396 ms · 2026-05-15T13:29:58.077325+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Let Γ_n, for n≥2, be the graph... Each knot is represented by a vertex... A directed edge exists... if K n→K′.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

[1]

On knot adjacency

Nikos Askitas and Efstratia Kalfagianni. “On knot adjacency”. In:Topol- ogy Appl.126.1-2 (2002), pp. 63–81.issn: 0166-8641,1879-3207.doi:10 . 1016/S0166-8641(02)00035-4.url:https://doi.org/10.1016/S0166- 8641(02)00035-4. 8 REFERENCES

work page doi:10.1016/s0166- 2002
[2]

Cosmetic crossings and Seifert matrices

Cheryl Balm et al. “Cosmetic crossings and Seifert matrices”. In:Comm. Anal. Geom.20.2 (2012), pp. 235–253.issn: 1019-8385,1944-9992.doi:10. 4310/CAG.2012.v20.n2.a1.url:https://doi.org/10.4310/CAG.2012. v20.n2.a1

work page doi:10.4310/cag.2012 2012
[3]

Knots without cosmetic cross- ings

Cheryl Jaeger Balm and Efstratia Kalfagianni. “Knots without cosmetic cross- ings”. In:Topology Appl.207 (2016), pp. 33–42.issn: 0166-8641,1879-3207. doi:10.1016/j.topol.2016.04.009.url:https://doi.org/10.1016/j. topol.2016.04.009

work page doi:10.1016/j.topol.2016.04.009.url:https://doi.org/10.1016/j 2016
[4]

Constructing and cataloging 2-adjacent knots

John Carney and Everett Meike. “Constructing and cataloging 2-adjacent knots”. In: ().url:https://arxiv.org/abs/2510.00291

work page arXiv
[5]

Stronglyn-trivial knots

Hugh Howards and John Luecke. “Stronglyn-trivial knots”. In:Bull. Lon- don Math. Soc.34.4 (2002), pp. 431–437.issn: 0024-6093,1469-2120.doi:10. 1112/S0024609301008980.url:https://doi.org/10.1112/S0024609301008980

work page doi:10.1112/s0024609301008980 2002
[6]

An obstruction of Gordian distance one and cosmetic crossings for genus one knots

Tetsuya Ito. “An obstruction of Gordian distance one and cosmetic crossings for genus one knots”. In:New York J. Math.28 (2022), pp. 175–181.issn: 1076-9803

work page 2022
[7]

Cosmetic crossing changes of fibered knots

Efstratia Kalfagianni. “Cosmetic crossing changes of fibered knots”. In:J. Reine Angew. Math.669 (2012), pp. 151–164.issn: 0075-4102,1435-5345.doi: 10.1515/crelle.2011.148.url:https://doi.org/10.1515/crelle. 2011.148

work page doi:10.1515/crelle.2011.148.url:https://doi.org/10.1515/crelle 2012
[8]

Knot adjacency and fibering

Efstratia Kalfagianni and Xiao-Song Lin. “Knot adjacency and fibering”. In: Trans. Amer. Math. Soc.360.6 (2008), pp. 3249–3261.issn: 0002-9947,1088- 6850.doi:10.1090/S0002-9947-08-04358-4.url:https://doi.org/10. 1090/S0002-9947-08-04358-4

work page doi:10.1090/s0002-9947-08-04358-4.url:https://doi.org/10 2008
[9]

Knot adjacency, genus and essen- tial tori

Efstratia Kalfagianni and Xiao-Song Lin. “Knot adjacency, genus and essen- tial tori”. In:Pacific J. Math.228.2 (2006), pp. 251–275.issn: 0030-8730,1945- 5844.doi:10.2140/pjm.2006.228.251.url:https://doi.org/10.2140/ pjm.2006.228.251

work page doi:10.2140/pjm.2006.228.251.url:https://doi.org/10.2140/ 2006
[10]

Problems in low-dimensional topology

“Problems in low-dimensional topology”. In:Geometric topology (Athens, GA, 1993). Ed. by Rob Kirby. Vol. 2.2. AMS/IP Stud. Adv. Math. Amer. Math. Soc., Providence, RI, 1997, pp. 35–473.isbn: 0-8218-0653-X.doi:10. 1090/amsip/002.2/02.url:https://doi.org/10.1090/amsip/002.2/02

work page doi:10.1090/amsip/002.2/02 1993
[11]

Cosmetic surgery in L-spaces and nu- gatory crossings

Tye Lidman and Allison H. Moore. “Cosmetic surgery in L-spaces and nu- gatory crossings”. In:Trans. Amer. Math. Soc.369.5 (2017), pp. 3639–3654. issn: 0002-9947,1088-6850.doi:10.1090/tran/6839.url:https://doi. org/10.1090/tran/6839

work page doi:10.1090/tran/6839.url:https://doi 2017
[12]

Link genus and the Conway moves

Martin Scharlemann and Abigail Thompson. “Link genus and the Conway moves”. In:Comment. Math. Helv.64.4 (1989), pp. 527–535.issn: 0010- 2571,1420-8946.doi:10 . 1007 / BF02564693.url:https : / / doi . org / 10 . 1007/BF02564693

work page 1989
[13]

2-adjacency between knots

Zhi-Xiong Tao. “2-adjacency between knots”. In:J. Knot Theory Ramifica- tions24.11 (2015), pp. 1550054, 18.issn: 0218-2165,1793-6527.doi:10.1142/ S0218216515500546.url:https://doi.org/10.1142/S0218216515500546

work page doi:10.1142/s0218216515500546 2015
[14]

A remark on 2-adjacency relation between Montesinos knots and 2-bridge knots

Ichiro Torisu. “A remark on 2-adjacency relation between Montesinos knots and 2-bridge knots”. In:Topology Appl.198 (2016), pp. 34–37.issn: 0166- 8641,1879-3207.doi:10.1016/j.topol.2015.10.012.url:https://doi. org/10.1016/j.topol.2015.10.012. REFERENCES 9

work page doi:10.1016/j.topol.2015.10.012.url:https://doi 2016
[15]

On 2-adjacency relation of two-bridge knots and links

Ichiro Torisu. “On 2-adjacency relation of two-bridge knots and links”. In: J. Aust. Math. Soc.84.1 (2008), pp. 139–144.issn: 1446-7887,1446-8107. doi:10 . 1017 / S1446788708000116.url:https : / / doi . org / 10 . 1017 / S1446788708000116

work page 2008
[16]

On nugatory crossings for knots

Ichiro Torisu. “On nugatory crossings for knots”. In:Topology Appl.92.2 (1999), pp. 119–129.issn: 0166-8641,1879-3207.doi:10.1016/S0166-8641(97) 00238-1.url:https://doi.org/10.1016/S0166-8641(97)00238-1. Marion Campisi San Jos´e State University San Jos´e, CA USA Email address:marion.campisi@sjsu.edu Brandy Doleshal Sam Houston State University Huntsville...

work page doi:10.1016/s0166-8641(97 1999