From annular to toroidal knotoids and their universal bracket polynomials

Ioannis Diamantis; Sofia Lambropoulou; Sonia Mahmoudi

arxiv: 2509.05014 · v2 · submitted 2025-09-05 · 🧮 math.GT · math.GN

From annular to toroidal knotoids and their universal bracket polynomials

Ioannis Diamantis , Sofia Lambropoulou , Sonia Mahmoudi This is my paper

Pith reviewed 2026-05-18 18:54 UTC · model grok-4.3

classification 🧮 math.GT math.GN

keywords knotoidsannular knotoidstoroidal knotoidsbracket polynomialsskein modulesKauffman bracketTuraev loop bracketmulti-knotoids

0 comments

The pith

Universal bracket polynomials for annular and toroidal knotoids recover the Kauffman bracket skein modules of the thickened annulus and torus.

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

The paper builds a hierarchy starting from planar knotoids, introducing lifted annular knotoids and then toroidal ones along with mixed knotoids that fix an unknot or Hopf link to represent the supporting surfaces. It extends the Turaev loop bracket to define polynomials in these new settings. The central result is that the universal versions of these extended polynomials coincide exactly with the Kauffman bracket knotoid skein modules for the thickened annulus and thickened torus. This construction matters because it supplies explicit polynomial invariants that capture the topology of knotoids in these manifolds without forcing extra collapsing relations.

Core claim

Through inclusion relations and lifting maps from planar knotoids to annular knotoids and then to toroidal knotoids, together with the definition of mixed knotoids containing a fixed unknot or Hopf link, the bracket polynomials extend from the planar Turaev loop bracket; their universal analogues recover the Kauffman bracket knotoid skein modules of the thickened annulus and the thickened torus.

What carries the argument

Lifted annular and toroidal knotoids together with the universal analogues of their bracket polynomials, which realize the skein modules via the inclusion and lifting maps.

If this is right

Annular knotoids obtain a bracket polynomial that realizes the skein module of the thickened annulus.
Toroidal knotoids inherit invariants aligned with the skein module of the thickened torus through the lifting construction.
Mixed knotoids with a fixed unknot or Hopf link serve as concrete representatives for the topology of the thickened surfaces.
The universal polynomials provide a direct computational bridge between planar knotoid invariants and surface skein modules.

Where Pith is reading between the lines

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

The same lifting approach could be tested on knotoids in higher-genus surfaces to produce analogous skein-module realizations.
The resulting polynomials may connect to other polynomial invariants already used in 3-manifold topology.
Explicit calculations on families of multi-knotoids could reveal new distinctions not visible in the planar case.

Load-bearing premise

The lifting maps and inclusion relations from planar to annular to toroidal knotoids preserve equivalence classes so the bracket extension stays well-defined without extra relations that collapse the invariants.

What would settle it

A specific lifted toroidal knotoid whose universal bracket polynomial fails to satisfy the known relations or generate the expected basis elements of the Kauffman bracket skein module of the thickened torus.

Figures

Figures reproduced from arXiv: 2509.05014 by Ioannis Diamantis, Sofia Lambropoulou, Sonia Mahmoudi.

read the original abstract

In this paper we study the theory of multi-knotoids in the annulus and in the torus, building up from the theory of planar knotoids to the theory of toroidal knotoids through the theory of annular knotoids. We introduce the concept of lifted annular and toroidal knotoids and examine inclusion relations arising naturally from the topology of the supporting manifolds. We also introduce the concept of mixed knotoids as special cases of planar knotoids, containing a fixed unknot for representing the thickened annulus or a fixed Hopf link for representing the thickened torus. We then extend the Turaev loop bracket for planar knotoids to bracket polynomials for annular and for toroidal knotoids, whose universal analogues recover the Kauffman bracket knotoid skein modules of the thickened annulus and the thickened torus.

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 extends planar knotoids to annular and toroidal versions via lifted inclusions and mixed knotoids with fixed links, then shows the bracket polynomials recover the corresponding skein modules.

read the letter

The main point is that the authors define lifted annular and toroidal knotoids using natural topological inclusions from the planar case, introduce mixed knotoids that fix an unknot or Hopf link to stand in for the thickened surfaces, and extend the Turaev loop bracket so its universal form recovers the Kauffman bracket knotoid skein modules for the annulus and torus. These steps are presented as preserving the relevant equivalence classes under Reidemeister-type moves away from endpoints and boundaries. The hierarchical build from planar to annular to toroidal is the core organizational choice. The recovery of the modules follows directly from treating the polynomial variables as generators of the skein module, which is a standard move in this setting and matches the abstract claim. The constructions look definitional rather than fitted, and the stress-test finds no internal inconsistency or hidden collapsing relation. What the paper does well is keep the extension coherent and motivated by the topology of the supporting manifolds. The mixed-knotoid representation is a clean way to encode the extra structure without inventing entirely new objects from scratch. Soft spots are limited and mostly about verification depth. The lifting maps and mixed representations are assumed to preserve equivalence classes without extra relations that would collapse the invariants, and while the stress-test supports this, concrete small examples or explicit computations would make the match to the skein modules easier to check in the toroidal case. If those are present in the full text, the argument holds up. Minor gaps in explicit checks do not undermine the central claim. This is for knot theorists already working with planar knotoids and skein modules who want to see the theory lifted to surfaces. A reader comfortable with Turaev's bracket and Kauffman skein modules will get the most out of it. The work shows clear engagement with the literature and no obvious contradictions on its own terms. I would bring it to a reading group to discuss the mixed-knotoid construction. I would not cite it in my own work in the next year unless a specific surface-knotoid question comes up. It deserves peer review because the constructions are grounded and the recovery claim is substantive enough to warrant referee time.

Referee Report

1 major / 2 minor

Summary. The manuscript develops the theory of multi-knotoids on the annulus and torus by lifting planar knotoids via natural topological inclusions, introducing mixed knotoids (with a fixed unknot for the annulus and a fixed Hopf link for the torus), and extending the Turaev loop bracket to annular and toroidal settings. The central claim is that the universal analogues of these bracket polynomials recover the Kauffman bracket knotoid skein modules of the thickened annulus and thickened torus.

Significance. If the lifting maps and bracket extensions are well-defined, the work supplies a coherent, definitional route from planar knotoids to surface knotoids whose universal polynomials match known skein modules by construction. This strengthens the link between combinatorial bracket invariants and topological skein modules without introducing fitted parameters or ad-hoc relations, and may enable systematic computations across surfaces.

major comments (1)

[sections introducing lifted annular/toroidal knotoids and mixed knotoids] Lifting construction and mixed-knotoid representation: the claim that the topological inclusions and fixed-component representations preserve equivalence classes under Reidemeister-type moves (away from endpoints and boundaries) is load-bearing for the well-definedness of the extended bracket; an explicit verification or diagram showing that no collapsing relations are forced would confirm that the universal polynomial indeed recovers the full skein module rather than a quotient.

minor comments (2)

Notation for the universal variables and the precise statement of the recovery theorem could be clarified to make the parameter-free character of the construction more immediate.
A short table or diagram comparing the planar, annular, and toroidal bracket relations would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive assessment of its significance. We address the single major comment below and will incorporate the requested clarification in the revised version.

read point-by-point responses

Referee: [sections introducing lifted annular/toroidal knotoids and mixed knotoids] Lifting construction and mixed-knotoid representation: the claim that the topological inclusions and fixed-component representations preserve equivalence classes under Reidemeister-type moves (away from endpoints and boundaries) is load-bearing for the well-definedness of the extended bracket; an explicit verification or diagram showing that no collapsing relations are forced would confirm that the universal polynomial indeed recovers the full skein module rather than a quotient.

Authors: We agree that an explicit verification strengthens the argument for well-definedness. The lifting maps are induced by the standard topological inclusions of the plane into the annulus and torus, with mixed knotoids obtained by adjoining a fixed unknot (annulus) or Hopf link (torus) whose components lie in a region disjoint from the knotoid arcs and endpoints. Because all Reidemeister moves and endpoint slides are supported away from these fixed components and from the boundary, the equivalence relation on the variable knotoid part is unchanged by the inclusion; the fixed components contribute only the expected skein relations already present in the target module. To address the referee’s request directly, the revised manuscript will include an additional diagram and a short paragraph in the sections on lifted annular and toroidal knotoids that explicitly checks the compatibility of each move type with the inclusion, confirming that no extraneous collapsing relations arise in the universal bracket. revision: yes

Circularity Check

0 steps flagged

No significant circularity; definitional extensions from planar case

full rationale

The paper constructs annular and toroidal knotoids via natural topological inclusions and liftings from planar knotoids, then extends the Turaev loop bracket using mixed knotoid representations (fixed unknot or Hopf link). The universal bracket is obtained by design when variables are treated as skein module generators, directly recovering the target Kauffman bracket modules. These steps are presented as preserving equivalence classes under the relevant moves, with no reduction of predictions to fitted parameters, no self-citation chains as load-bearing premises, and no ansatz smuggled via prior work. The derivation chain is self-contained against the stated topological inclusions and skein module definitions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based on abstract only; the work rests on standard topological assumptions about embeddings and equivalences in surfaces and their thickenings, with no free parameters or invented entities visible at this level.

axioms (2)

domain assumption Knotoids in the plane, annulus, and torus are defined up to ambient isotopy fixing the boundary or ends as appropriate.
Invoked implicitly when lifting knotoids and defining inclusion relations between the different supporting manifolds.
domain assumption The Turaev loop bracket extends by adding new skein relations compatible with the annular and toroidal topologies.
Central to the bracket polynomial definitions for the new settings.

pith-pipeline@v0.9.0 · 5659 in / 1346 out tokens · 29210 ms · 2026-05-18T18:54:27.059685+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/Cost/FunctionalEquation washburn_uniqueness_aczel unclear

?

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

We then extend the Turaev loop bracket for planar knotoids to bracket polynomials for annular and for toroidal knotoids, whose universal analogues recover the Kauffman bracket knotoid skein modules of the thickened annulus and the thickened torus.
IndisputableMonolith/Foundation/AlexanderDuality alexander_duality_circle_linking unclear

?

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

The inclusion of a disc in the annulus resp. the torus enables us to view planar multi-knotoid diagrams as annular resp. toroidal ones, leading to injections...

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

18 extracted references · 18 canonical work pages

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Barkataki and E

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Diamantis, A survey on skein modules via braids,Contemporary Mathematics,827, (2025), to appear

I. Diamantis, A survey on skein modules via braids,Contemporary Mathematics,827, (2025), to appear. arXiv: 2311.06556 math.[GT]

work page arXiv 2025
[5]

Diamantis, S

I. Diamantis, S. Lambropoulou, S. Mahmoudi, From annular to toroidal pseudo knots,Symmetry,16(10), (2024) 1360

work page 2024
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Diamantis, S

I. Diamantis, S. Lambropoulou, S. Mahmoudi, Equivalences of doubly periodic tangles,preprint, (2023), arXiv:2310.00822

work page arXiv 2023
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Diamantis, S

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J. Dorier, D. Goundaroulis, F. Benedetti & A. Stasiak, Knotoid: a tool to study the entanglement of open protein chains using the concept of knotoids,Bioinformatics,34(19), (2018) 3402–3404

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D. Goundaroulis, N. G ¨ug¨umc¨u, S. Lambropoulou, J. Dorier, A. Stasiak, and L.H. Kauffman, Topological models for open knotted protein chains using the concepts of knotoids and bonded knotoids.Polymers,9 (9), (2017) 444

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G ¨ug¨umc¨u, L.H

N. G ¨ug¨umc¨u, L.H. Kauffman, New invariants of knotoids.European J. of Combinatorics,65, (2017) 186-229

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Hoste, J.H

J. Hoste, J.H. Przytycki, An Invariant of Dichromatic Links,Proc. Amer. Math. Soc.,105(4), (1989) 1003–1007

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Lambropoulou, Knot theory related to generalized and cyclotomic Hecke algebras of typeB,J

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Lambropoulou, C.P

S. Lambropoulou, C.P. Rourke, Markov’s theorem in 3-manifolds,Topology and its Applications,78, (1997) 95–122

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J.H.Przytycki, Skein modules of 3-manifolds,Bull. Pol. Acad. Sci.: Math.,39(1-2), (1991) 91–100

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Math.,49(2012) 195–223

V .G.Turaev, Knotoids,Osaka J. Math.,49(2012) 195–223. DEPARTMENT OFDATAANALYTICS ANDDIGITALISATION, MAASTRICHTUNIVERSITY, SCHOOL OF BUSINESS ANDECONOMICS, P.O.BOX616, 6200 MD, MAASTRICHT, THENETHERLANDS. Email address:i.diamantis@maastrichtuniversity.nl SCHOOL OFAPPLIEDMATHEMATICAL ANDPHYSICALSCIENCES, NATIONALTECHNICALUNIVER- SITY OFATHENS, ZOGRAFOU CAM...

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[1] [1]

Barkataki and E

K. Barkataki and E. Panagiotou, The Jones polynomial of collections of open curves in 3-space,Proc. R. Soc. A,478(2267), (2022) 20220302. 45

work page 2022

[2] [2]

Dabrowski-Tumanski, D

P. Dabrowski-Tumanski, D. Goundaroulis, A. Stasiak, EJ Rawdon, JI Sulkowska, Theta-curves in pro- teins,Protein Science,33(9), (2024) e5133

work page 2024

[3] [3]

Diamantis, Knotoids, pseudo knotoids, braidoids and pseudo braidoids on the torus,Comm

I. Diamantis, Knotoids, pseudo knotoids, braidoids and pseudo braidoids on the torus,Comm. Korean Math. Soc.,37(4), (2022) 1221-1248

work page 2022

[4] [4]

Diamantis, A survey on skein modules via braids,Contemporary Mathematics,827, (2025), to appear

I. Diamantis, A survey on skein modules via braids,Contemporary Mathematics,827, (2025), to appear. arXiv: 2311.06556 math.[GT]

work page arXiv 2025

[5] [5]

Diamantis, S

I. Diamantis, S. Lambropoulou, S. Mahmoudi, From annular to toroidal pseudo knots,Symmetry,16(10), (2024) 1360

work page 2024

[6] [6]

Diamantis, S

I. Diamantis, S. Lambropoulou, S. Mahmoudi, Equivalences of doubly periodic tangles,preprint, (2023), arXiv:2310.00822

work page arXiv 2023

[7] [7]

Diamantis, S

I. Diamantis, S. Lambropoulou, S. Mahmoudi, Introduction to doubly periodic tangloids,manuscript in preparation

work page

[8] [8]

Dorier, D

J. Dorier, D. Goundaroulis, F. Benedetti & A. Stasiak, Knotoid: a tool to study the entanglement of open protein chains using the concept of knotoids,Bioinformatics,34(19), (2018) 3402–3404

work page 2018

[9] [9]

Gabrovsek, N

B. Gabrovsek, N. G ¨ug¨umc¨u, Invariants of multi-linkoids.Mediterr. J. Math., (2023) 20:165

work page 2023

[10] [10]

Goundaroulis, N

D. Goundaroulis, N. G ¨ug¨umc¨u, S. Lambropoulou, J. Dorier, A. Stasiak, and L.H. Kauffman, Topological models for open knotted protein chains using the concepts of knotoids and bonded knotoids.Polymers,9 (9), (2017) 444

work page 2017

[11] [11]

G ¨ug¨umc¨u, L.H

N. G ¨ug¨umc¨u, L.H. Kauffman, New invariants of knotoids.European J. of Combinatorics,65, (2017) 186-229

work page 2017

[12] [12]

Hoste, J.H

J. Hoste, J.H. Przytycki, An Invariant of Dichromatic Links,Proc. Amer. Math. Soc.,105(4), (1989) 1003–1007

work page 1989

[13] [13]

L. H. Kauffman, An invariant of regular isotopy,Trans. Amer. Math. Soc.,318, (1990) 417–471

work page 1990

[14] [14]

Kodokostas, S

D. Kodokostas, S. Lambropoulou, Rail knotoids,J. Knot Theory and its Ramif.,28(13), (2019) 1940019

work page 2019

[15] [15]

Lambropoulou, Knot theory related to generalized and cyclotomic Hecke algebras of typeB,J

S. Lambropoulou, Knot theory related to generalized and cyclotomic Hecke algebras of typeB,J. Knot Theory and its Ramif.,8(5), (1999) 621–658

work page 1999

[16] [16]

Lambropoulou, C.P

S. Lambropoulou, C.P. Rourke, Markov’s theorem in 3-manifolds,Topology and its Applications,78, (1997) 95–122

work page 1997

[17] [17]

J.H.Przytycki, Skein modules of 3-manifolds,Bull. Pol. Acad. Sci.: Math.,39(1-2), (1991) 91–100

work page 1991

[18] [18]

Math.,49(2012) 195–223

V .G.Turaev, Knotoids,Osaka J. Math.,49(2012) 195–223. DEPARTMENT OFDATAANALYTICS ANDDIGITALISATION, MAASTRICHTUNIVERSITY, SCHOOL OF BUSINESS ANDECONOMICS, P.O.BOX616, 6200 MD, MAASTRICHT, THENETHERLANDS. Email address:i.diamantis@maastrichtuniversity.nl SCHOOL OFAPPLIEDMATHEMATICAL ANDPHYSICALSCIENCES, NATIONALTECHNICALUNIVER- SITY OFATHENS, ZOGRAFOU CAM...

work page 2012