pith. sign in

arxiv: 2510.12372 · v3 · submitted 2025-10-14 · 🧮 math.GT

A preorder on the set of links with applications to symmetric unions

Michel Boileau , Teruaki Kitano , Yuta Nozaki This is my paper

Pith reviewed 2026-05-18 07:53 UTC · model grok-4.3

classification 🧮 math.GT MSC 57M25
keywords preorderorbifold groupMontesinos linksymmetric unionepimorphism2-bridge linkdeterminantsmall link
0
0 comments X

The pith

A preorder on links induced by π-orbifold group epimorphisms constrains Montesinos links and identifies knots without symmetric union presentations.

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

This paper defines a preorder on the set of links in the 3-sphere using epimorphisms between their π-orbifold groups that fit a commutative diagram with the link groups. The central results show how this relation restricts the possible images of Montesinos links with several rational tangles to simpler Montesinos links or connected sums. It establishes finiteness for the number of links below any small link and shows that determinant-zero links precede all 2-bridge links. These properties lead to a criterion that can prove a knot does not have a symmetric union presentation. A reader would care because this gives an algebraic method to relate and distinguish links based on their group structures.

Core claim

The authors introduce the relation L ≽ L' on links when there is an epimorphism G^orb(L) to G^orb(L') fitting into a commutative diagram. They prove that if L is a Montesinos link with r rational tangles for r at least 3 and L ≽ L', then L' is a Montesinos link with at most r+1 rational tangles or a certain connected sum. If L is small, there are only finitely many such L'. If the determinant of L is zero, then L ≽ L' for every 2-bridge link L'. This preorder provides a criterion to show that a given knot does not admit a symmetric union presentation.

What carries the argument

the preorder on links given by epimorphisms of π-orbifold groups fitting a commutative diagram

If this is right

  • If L is Montesinos with r≥3 rational tangles and L ≽ L', then L' is Montesinos with ≤r+1 tangles or a connected sum.
  • Small links precede only finitely many links in the preorder.
  • Determinant-zero links precede every 2-bridge link.
  • The preorder gives a criterion to show certain knots lack symmetric union presentations.

Where Pith is reading between the lines

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

  • This preorder might allow systematic comparison of link complexities across different families.
  • The finiteness property could enable algorithmic checks for small links in knot censuses.
  • The symmetric union criterion may be combined with other invariants to classify knots that do admit such presentations.
  • Similar relations could be defined using other quotients of link groups for different applications.

Load-bearing premise

The definition of the preorder via an epimorphism of orbifold groups fitting exactly the specified commutative diagram is the appropriate one to yield the stated structural properties.

What would settle it

A Montesinos link with three rational tangles that precedes a link neither Montesinos with at most four tangles nor a connected sum would falsify the main theorem; likewise, a small link preceding infinitely many links or a knot with symmetric union that the criterion flags as impossible.

Figures

Figures reproduced from arXiv: 2510.12372 by Michel Boileau, Teruaki Kitano, Yuta Nozaki.

Figure 1
Figure 1. Figure 1: Symmetric union (D∪D∗ )(∞, n1, . . . , nk) and its partial knot KD. The symmetric union construction is not unique and the isotopy type of K = (D ∪ D∗ )(∞, n1, . . . , nk) depends on both the diagram D and the loca￾tion of the tangle replacements. When there is a single tangle replacement, the construction is due to Kinoshita and Terasaka [40]. The extension to multiple symmetric tangle replacements is due… view at source ↗
read the original abstract

For a link $L$ in the $3$-sphere, the $\pi$-orbifold group $G^\mathrm{orb}(L)$ is defined as a quotient of the link group $G(L)$ of $L$. When there exists an epimorphism $G^\mathrm{orb}(L)\to G^\mathrm{orb}(L')$ fitting into a certain commutative diagram, we define a relation $L\succeq L'$ and explore the relationships between the two links. Specifically, we prove that if $L\succeq L'$ and $L$ is a Montesinos link with $r$ rational tangles $(r\geq 3)$, then $L'$ is either a Montesinos link with at most $r+1$ rational tangles or a certain connected sum. We further show that if $L$ is a small link, then there are only finitely many links $L'$ satisfying $L\succeq L'$. In contrast, if $L$ has determinant zero, then $L\succeq L'$ for every $2$-bridge link $L'$. Our main applications concern symmetric unions of knots. In particular, we provide a criterion showing that a given knot does not admit a symmetric union presentation.

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

2 major / 2 minor

Summary. The manuscript defines a preorder ≽ on links in S³ by the existence of an epimorphism G^orb(L) → G^orb(L') of π-orbifold groups that fits into a specified commutative diagram. It proves that if L ≽ L' and L is Montesinos with r ≥ 3 rational tangles then L' is Montesinos with at most r+1 tangles or a certain connected sum; that small links L admit only finitely many L' with L ≽ L'; that det(L)=0 implies L ≽ L' for every 2-bridge link L'; and that the preorder yields an obstruction criterion for knots admitting symmetric-union presentations.

Significance. If the central claims hold, the construction supplies a new, diagram-constrained preorder on links that directly yields structural restrictions on Montesinos and small links together with a concrete obstruction for symmetric unions. The finiteness statement for small links and the explicit bound on the number of tangles for Montesinos links are potentially useful for classification problems in knot theory.

major comments (2)
  1. [§3] §3 (definition of ≽): the claim that the diagram condition induces a preorder requires an explicit verification that the relation is transitive; the manuscript should record the composition of the two epimorphisms and confirm that the resulting diagram still commutes.
  2. [Theorem 4.2] Theorem 4.2 (Montesinos case): the bound 'at most r+1 rational tangles' is derived from the epimorphism on orbifold groups; the argument should state precisely which property of the peripheral subgroups or the tangle decomposition is used to obtain the +1 increment rather than a stricter bound.
minor comments (2)
  1. [Definition 2.3] The commutative diagram in Definition 2.3 is central; a single displayed diagram with all maps labeled would improve readability.
  2. [Introduction] Notation: G^orb(L) is introduced as a quotient of G(L); a one-sentence reminder of the precise quotient (meridian-to-order-2) in the introduction would help readers who are not specialists in orbifold groups.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation and the precise suggestions that help clarify the manuscript. We address each major comment below.

read point-by-point responses

  1. Referee: [§3] §3 (definition of ≽): the claim that the diagram condition induces a preorder requires an explicit verification that the relation is transitive; the manuscript should record the composition of the two epimorphisms and confirm that the resulting diagram still commutes.

    Authors: We agree that an explicit check of transitivity is required. In the revised version we have added a short paragraph in §3 that composes the two epimorphisms φ : G^orb(L) → G^orb(M) and ψ : G^orb(M) → G^orb(L') and verifies commutativity of the resulting diagram by chasing the two squares and using the naturality of the maps from the link groups to the orbifold groups. revision: yes

  2. Referee: [Theorem 4.2] Theorem 4.2 (Montesinos case): the bound 'at most r+1 rational tangles' is derived from the epimorphism on orbifold groups; the argument should state precisely which property of the peripheral subgroups or the tangle decomposition is used to obtain the +1 increment rather than a stricter bound.

    Authors: We thank the referee for this request for precision. The +1 increment arises because the epimorphism may send two distinct peripheral subgroups (corresponding to consecutive rational tangles) to the same subgroup in the target orbifold group. We have expanded the proof of Theorem 4.2 to identify this merging property of peripheral subgroups explicitly and to explain why a stricter bound does not follow in general. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper explicitly defines the preorder via existence of an epimorphism of orbifold groups fitting a specified commutative diagram, then derives all structural results (Montesinos restrictions, finiteness for small links, determinant-zero behavior, and symmetric-union criterion) from that definition together with standard facts about link groups and quotients. No self-referential definitions appear, no fitted parameters are renamed as predictions, and no load-bearing claims reduce to self-citations or prior ansatzes by the authors. The work is a pure-mathematical development of a new relation whose consequences follow independently once the diagram condition is granted.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper works entirely within the standard framework of 3-manifold fundamental groups and orbifold quotients; no new entities are postulated and no numerical parameters are fitted.

axioms (2)
  • domain assumption The π-orbifold group G^orb(L) is defined as a quotient of the link group G(L).
    Stated in the opening sentence of the abstract as the starting point for the preorder.
  • ad hoc to paper Epimorphisms of orbifold groups that fit the commutative diagram induce the preorder relation ≽ on links.
    This is the definition that generates all later theorems; it is introduced specifically for this work.

pith-pipeline@v0.9.0 · 5760 in / 1488 out tokens · 42793 ms · 2026-05-18T07:53:27.954494+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

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

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

66 extracted references · 66 canonical work pages

  1. [1]

    Problems in low-dimensional topology. In R. Kirby, editor,Geometric topology (Athens, GA, 1993), volume 2 ofAMS/IP Stud. Adv. Math., pages 35–473. Amer. Math. Soc., Providence, RI, 1997

    work page 1993

  2. [2]

    Baader, R

    S. Baader, R. Blair, and A. Kjuchukova. Coxeter groups and meridional rank of links. Math. Ann., 379(3-4):1533–1551, 2021

    work page 2021

  3. [3]

    Baader, R

    S. Baader, R. Blair, A. Kjuchukova, and F. Misev. The bridge number of arborescent links with many twigs.Algebr. Geom. Topol., 23(1):75–85, 2023

    work page 2023

  4. [4]

    Boileau, S

    M. Boileau, S. Boyer, A. W. Reid, and S. Wang. Simon’s conjecture for two-bridge knots.Comm. Anal. Geom., 18(1):121–143, 2010

    work page 2010

  5. [5]

    Boileau, T

    M. Boileau, T. Kitano, and Y. Nozaki. On the genera of symmetric unions of knots. arXiv:2407.0988v2, 2025

    work page arXiv 2025

  6. [6]

    Boileau, B

    M. Boileau, B. Leeb, and J. Porti. Geometrization of 3-dimensional orbifolds.Ann. of Math. (2), 162(1):195–290, 2005

    work page 2005

  7. [7]

    Boileau, S

    M. Boileau, S. Maillot, and J. Porti.Three-dimensional orbifolds and their geometric structures, volume 15 ofPanoramas et Synth` eses [Panoramas and Syntheses]. Soci´ et´ e Math´ ematique de France, Paris, 2003

    work page 2003

  8. [8]

    Boileau and J.-P

    M. Boileau and J.-P. Otal. Scindements de Heegaard et groupe des hom´ eotopies des petites vari´ et´ es de Seifert.Invent. Math., 106(1):85–107, 1991

    work page 1991

  9. [9]

    Boileau and J

    M. Boileau and J. Porti. Geometrization of 3-orbifolds of cyclic type.Ast´ erisque, (272):208, 2001. Appendix A by Michael Heusener and Porti

    work page 2001

  10. [10]

    Boileau and H

    M. Boileau and H. Zieschang. Heegaard genus of closed orientable Seifert 3-manifolds. Invent. Math., 76(3):455–468, 1984

    work page 1984

  11. [11]

    Boileau and H

    M. Boileau and H. Zieschang. Nombre de ponts et g´ en´ erateurs m´ eridiens des entrelacs de Montesinos.Comment. Math. Helv., 60(2):270–279, 1985

    work page 1985

  12. [12]

    Boileau and B

    M. Boileau and B. Zimmermann. Theπ-orbifold group of a link.Math. Z., 200(2):187– 208, 1989

    work page 1989

  13. [13]

    Bonahon and L

    F. Bonahon and L. Siebenmann. The classification of Seifert fibred 3-orbifolds. In Low-dimensional topology (Chelwood Gate, 1982), volume 95 ofLondon Math. Soc. Lecture Note Ser., pages 19–85. Cambridge Univ. Press, Cambridge, 1985

    work page 1982

  14. [14]

    Bonahon and L

    F. Bonahon and L. Siebenmann. New Geometric Splittings of Classical Knots and the Classification and Symmetries of Arborescent knots, 2016

    work page 2016

  15. [15]

    Boyer and X

    S. Boyer and X. Zhang. On Culler-Shalen seminorms and Dehn filling.Ann. of Math. (2), 148(3):737–801, 1998

    work page 1998

  16. [16]

    Burde and K

    G. Burde and K. Murasugi. Links and Seifert fiber spaces.Duke Math. J., 37:89–93, 1970

    work page 1970

  17. [17]

    Burde, H

    G. Burde, H. Zieschang, and M. Heusener.Knots, volume 5 ofDe Gruyter Studies in Mathematics. De Gruyter, Berlin, extended edition, 2014

    work page 2014

  18. [18]

    Casson and D

    A. Casson and D. Jungreis. Convergence groups and Seifert fibered 3-manifolds.In- vent. Math., 118(3):441–456, 1994

    work page 1994

  19. [19]

    J. C. Cha and M. Suzuki. Non-meridional epimorphisms of knot groups.Algebr. Geom. Topol., 16(2):1135–1155, 2016. 38

    work page 2016

  20. [20]

    Cooper, C

    D. Cooper, C. D. Hodgson, and S. P. Kerckhoff.Three-dimensional orbifolds and cone-manifolds, volume 5 ofMSJ Memoirs. Mathematical Society of Japan, Tokyo,

    work page

  21. [21]

    With a postface by Sadayoshi Kojima

    work page

  22. [22]

    Culler, C

    M. Culler, C. M. Gordon, J. Luecke, and P. B. Shalen. Dehn surgery on knots.Ann. of Math. (2), 125(2):237–300, 1987

    work page 1987

  23. [23]

    Culler and P

    M. Culler and P. B. Shalen. Varieties of group representations and splittings of 3- manifolds.Ann. of Math. (2), 117(1):109–146, 1983

    work page 1983

  24. [24]

    W. D. Dunbar. Hierarchies for 3-orbifolds.Topology Appl., 29(3):267–283, 1988

    work page 1988

  25. [25]

    M. J. Dunwoody. An equivariant sphere theorem.Bull. London Math. Soc., 17(5):437– 448, 1985

    work page 1985

  26. [26]

    R. H. Fox. A note on branched cyclic covering of spheres.Rev. Mat. Hisp.-Amer. (4), 32:158–166, 1972

    work page 1972

  27. [27]

    S. Friedl. Centralizers in 3-manifold groups (twisted topological invariants and topol- ogy of low-dimensional manifolds).RIMS Kˆ okyˆ uroku, 1747, 6 2011

    work page 2011

  28. [28]

    D. Gabai. Convergence groups are Fuchsian groups.Ann. of Math. (2), 136(3):447– 510, 1992

    work page 1992

  29. [29]

    Gonz´ alez-Acu˜ na

    F. Gonz´ alez-Acu˜ na. Homomorphs of knot groups.Ann. of Math. (2), 102(2):373–377, 1975

    work page 1975

  30. [30]

    C. M. Gordon and R. A. Litherland. Incompressible surfaces in branched coverings. InThe Smith conjecture (New York, 1979), volume 112 ofPure Appl. Math., pages 139–152. Academic Press, Orlando, FL, 1984

    work page 1979

  31. [31]

    Groves, M

    D. Groves, M. Hull, and H. Liang. Homomorphisms to 3-manifold groups. arXiv:2103.06095v3, 2024

    work page arXiv 2024

  32. [32]

    Hashizume

    Y. Hashizume. On the uniqueness of the decomposition of a link.Osaka Math. J., 10:283–300, 1958

    work page 1958

  33. [33]

    J. Hempel. Residual finiteness for 3-manifolds. InCombinatorial group theory and topology (Alta, Utah, 1984), volume 111 ofAnn. of Math. Stud., pages 379–396. Princeton Univ. Press, Princeton, NJ, 1987

    work page 1984

  34. [34]

    Hempel.3-manifolds

    J. Hempel.3-manifolds. AMS Chelsea Publishing, Providence, RI, 2004. Reprint of the 1976 original

    work page 2004

  35. [35]

    Ichihara, K

    K. Ichihara, K. Ishikawa, E. Matsudo, and M. Suzuki. Two-tone colorings and sur- jective dihedral representations for links.Osaka J. Math., to appear

    work page

  36. [36]

    W. H. Jaco and P. B. Shalen. Seifert fibered spaces in 3-manifolds.Mem. Amer. Math. Soc., 21(220):viii+192, 1979

    work page 1979

  37. [37]

    Jankins and W

    M. Jankins and W. D. Neumann.Lectures on Seifert manifolds, volume 2 ofBrandeis Lecture Notes. Brandeis University, Waltham, MA, 1983

    work page 1983

  38. [38]

    D. Johnson. Homomorphs of knot groups.Proc. Amer. Math. Soc., 78(1):135–138, 1980

    work page 1980

  39. [39]

    Kawauchi

    A. Kawauchi. Almost identical imitations of (3,1)-dimensional manifold pairs.Osaka J. Math., 26(4):743–758, 1989

    work page 1989

  40. [40]

    Kawauchi, editor.A survey of knot theory

    A. Kawauchi, editor.A survey of knot theory. Birkh¨ auser Verlag, Basel, 1996. Trans- lated and revised from the 1990 Japanese original by the author

    work page 1996

  41. [41]

    Kinoshita and H

    S. Kinoshita and H. Terasaka. On unions of knots.Osaka Math. J., 9:131–153, 1957

    work page 1957

  42. [42]

    Kleiner and J

    B. Kleiner and J. Lott. Notes on Perelman’s papers.Geom. Topol., 12(5):2587–2855, 2008

    work page 2008

  43. [43]

    A. W. Knapp. Doubly generated Fuchsian groups.Michigan Math. J., 15:289–304, 1968

    work page 1968

  44. [44]

    C. Lamm. Symmetric unions and ribbon knots.Osaka J. Math., 37(3):537–550, 2000

    work page 2000

  45. [45]

    Lamm and T

    C. Lamm and T. Tanaka. On partial knots for symmetric unions.Topology Appl., 375:Paper No. 109593, 2025

    work page 2025

  46. [46]

    A. M. Macbeath. Geometrical realization of isomorphisms between plane groups.Bull. Amer. Math. Soc., 71:629–630, 1965. 39

    work page 1965

  47. [47]

    Maskit.Kleinian groups, volume 287 ofGrundlehren der mathematischen Wis- senschaften [Fundamental Principles of Mathematical Sciences]

    B. Maskit.Kleinian groups, volume 287 ofGrundlehren der mathematischen Wis- senschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 1988

    work page 1988

  48. [48]

    Meeks, III, L

    W. Meeks, III, L. Simon, and S. T. Yau. Embedded minimal surfaces, exotic spheres, and manifolds with positive Ricci curvature.Ann. of Math. (2), 116(3):621–659, 1982

    work page 1982

  49. [49]

    W. H. Meeks, III and P. Scott. Finite group actions on 3-manifolds.Invent. Math., 86(2):287–346, 1986

    work page 1986

  50. [50]

    J. M. Montesinos. Seifert manifolds that are ramified two-sheeted cyclic coverings. Bol. Soc. Mat. Mexicana (2), 18:1–32, 1973

    work page 1973

  51. [51]

    J. M. Montesinos.Classical tessellations and three-manifolds. Universitext. Springer- Verlag, Berlin, 1987

    work page 1987

  52. [52]

    Morgan and G

    J. Morgan and G. Tian.Ricci flow and the Poincar´ e conjecture, volume 3 ofClay Mathematics Monographs. American Mathematical Society, Providence, RI; Clay Mathematics Institute, Cambridge, MA, 2007

    work page 2007

  53. [53]

    J. W. Morgan and H. Bass, editors.The Smith conjecture, volume 112 ofPure and Applied Mathematics. Academic Press, Inc., Orlando, FL, 1984. Papers presented at the symposium held at Columbia University, New York, 1979

    work page 1984

  54. [54]

    W. D. Neumann and F. Raymond. Seifert manifolds, plumbing,µ-invariant and ori- entation reversing maps. InAlgebraic and geometric topology (Proc. Sympos., Univ. California, Santa Barbara, Calif., 1977), volume 664 ofLecture Notes in Math., pages 163–196. Springer, Berlin, 1978

    work page 1977

  55. [55]

    U. Oertel. Closed incompressible surfaces in complements of star links.Pacific J. Math., 111(1):209–230, 1984

    work page 1984

  56. [56]

    Orlik.Seifert manifolds

    P. Orlik.Seifert manifolds. Lecture Notes in Mathematics, Vol. 291. Springer-Verlag, Berlin-New York, 1972

    work page 1972

  57. [57]

    A. W. Reid and S. Wang. Non-Haken 3-manifolds are not large with respect to map- pings of non-zero degree.Comm. Anal. Geom., 7(1):105–132, 1999

    work page 1999

  58. [58]

    Y. W. Rong. Degree one maps between geometric 3-manifolds.Trans. Amer. Math. Soc., 332(1):411–436, 1992

    work page 1992

  59. [59]

    M. Sakuma. A survey of the impact of Thurston’s work on knot theory. InIn the tradition of Thurston—geometry and topology, pages 67–160. Springer, Cham, 2020

    work page 2020

  60. [60]

    Schubert

    H. Schubert. ¨Uber eine numerische Knoteninvariante.Math. Z., 61:245–288, 1954

    work page 1954

  61. [61]

    D. S. Silver, W. Whitten, and S. G. Williams. Knot groups with many killers.Bull. Aust. Math. Soc., 81(3):507–513, 2010

    work page 2010

  62. [62]

    J. Simon. Roots and centralizers of peripheral elements in knot groups.Math. Ann., 222(3):205–209, 1976

    work page 1976

  63. [63]

    T. Tanaka. The amphicheiral 2-bridge knots with symmetric union presentations. Bull. Korean Math. Soc., 61(2):421–431, 2024

    work page 2024

  64. [64]

    Waldhausen

    F. Waldhausen. Eine Klasse von 3-dimensionalen Mannigfaltigkeiten. I, II.Invent. Math., 3:308–333; ibid. 4 (1967), 87–117, 1967

    work page 1967

  65. [65]

    L. Watson. Surgery obstructions from Khovanov homology.Selecta Math. (N.S.), 18(2):417–472, 2012

    work page 2012

  66. [66]

    Yokoyama

    M. Yokoyama. A general theory of orbifolds, and essential 2-suborbifolds respecting graph products.Topology Appl., 207:62–95, 2016. 40 Aix Marseille University, CNRS, Centrale Marseille, I2M,, 3 Pl. Victor Hugo, 13003 Marseille, France Email address:michel.boileau@univ-amu.fr Department of Information Systems Science, F aculty of Science and Engi- neering...

    work page 2016