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arxiv: 2604.07790 · v1 · submitted 2026-04-09 · 🧮 math.GT

A Dehornoy-Type Ordering on Plat Presentation Classes

Makoto Ozawa This is my paper

Pith reviewed 2026-05-10 18:07 UTC · model grok-4.3

classification 🧮 math.GT
keywords Dehornoy orderplat presentationsHilden subgroupsbridge numberbraid groupslink theorycanonical representativesfiniteness conjecture
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The pith

The Dehornoy order on braids induces a strict total order on the double cosets of Hilden subgroups, yielding canonical plat presentations for links.

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

The paper shows how to impose a strict total order on the set of all 2n-plat presentation classes of links by transferring the Dehornoy ordering from the braid group after fixing a complexity function that selects representatives. For any link whose bridge number is at most n, the same construction orders all its n-bridge positions up to bridge isotopy. The author isolates a distinguished canonical class for each link and proves that the globally chosen Dehornoy canonical braid coincides with the representative picked inside the corresponding double coset. This machinery recasts the fixed-level bridge finiteness conjecture as a statement about boundedness of the canonical representatives. The viewpoint treats bridge positions as an ordered, finite-level model that could organize the larger space of geometric placements of a link.

Core claim

For each integer n ≥ 1, after fixing a proper complexity function on the braid group B_{2n}, the Dehornoy order defines a strict total order on the set P_{2n} = H_{2n} ∖ B_{2n} / H_{2n} of 2n-plat presentation classes. For a link type L with bridge number b(L) ≤ n this induces a strict total order on the subset P^{(n)}(L) of bridge isotopy classes of n-bridge positions of L. The paper defines a distinguished class CanPlat_D^{(n)}(L) and shows that the globally chosen Dehornoy canonical braid agrees with the cosetwise canonical representative of the associated Hilden double coset. The construction reformulates the fixed-level bridge finiteness conjecture in terms of boundedness of canonical 2

What carries the argument

The Dehornoy order on B_{2n} transferred to the double coset space P_{2n} via a proper complexity function that selects a unique minimal representative in each coset.

If this is right

  • Every link with bridge number at most n has its n-bridge isotopy classes strictly ordered.
  • A distinguished canonical plat class CanPlat_D^{(n)}(L) exists for each such link.
  • The Dehornoy canonical braid coincides with the cosetwise canonical representative.
  • The fixed-level bridge finiteness conjecture is equivalent to the boundedness of these canonical representatives.

Where Pith is reading between the lines

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

  • The total order could support systematic enumeration or algorithmic comparison of minimal bridge presentations.
  • Bridge positions become an ordered discrete set that might serve as a controlled approximation to arbitrary geometric embeddings of links.
  • The same ordering technique might be adaptable to other subgroups or presentations in knot theory to produce further canonical forms.

Load-bearing premise

There exists a proper complexity function on the braid group B_{2n} such that the Dehornoy order produces a strict total order on the double cosets without leaving any two distinct classes incomparable.

What would settle it

An explicit link together with two distinct n-bridge plat presentations whose associated braids have Dehornoy representatives that remain incomparable after the complexity function is applied, or whose chosen representatives fail to match the globally Dehornoy canonical braid.

read the original abstract

For each integer $n\ge 1$, after fixing a proper complexity function on the braid group $\B_{2n}$, we use the Dehornoy order to define a strict total order on the set \[ \mathcal P_{2n}=H_{2n}\backslash \B_{2n}/H_{2n} \] of $2n$--plat presentation classes. For a link type $\mathcal L$ with bridge number $b(\mathcal L)\le n$, this induces a strict total order on the subset $\mathcal P^{(n)}(\mathcal L)$ corresponding to bridge isotopy classes of $n$--bridge positions of $\mathcal L$. We also define a distinguished class $\CanPlat_D^{(n)}(\mathcal L)$ and show that the globally chosen Dehornoy canonical braid agrees with the cosetwise canonical representative of the associated Hilden double coset. As an application, we reformulate the fixed-level bridge finiteness conjecture in terms of boundedness of canonical representatives. This viewpoint supports the role of bridge positions as a structured finite-level model for studying the otherwise vast collection of geometric positions of a link.

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

0 major / 3 minor

Summary. For each integer n≥1, after fixing a proper complexity function on the braid group B_{2n}, the Dehornoy order is used to define a strict total order on the set P_{2n}=H_{2n}∖B_{2n}/H_{2n} of 2n-plat presentation classes. For a link type L with bridge number b(L)≤n, this induces a strict total order on the subset P^{(n)}(L) corresponding to bridge isotopy classes of n-bridge positions of L. The paper defines a distinguished class CanPlat_D^{(n)}(L) and shows that the globally chosen Dehornoy canonical braid agrees with the cosetwise canonical representative of the associated Hilden double coset. As an application, the fixed-level bridge finiteness conjecture is reformulated in terms of boundedness of canonical representatives, supporting bridge positions as a structured finite-level model for geometric positions of links.

Significance. If the construction holds, the work provides a concrete extension of the Dehornoy left-invariant total order from braids to the double-coset space of plat presentations, yielding a total order on bridge isotopy classes of links. This is significant for its direct reformulation of the bridge finiteness conjecture via boundedness of canonical representatives and for positioning bridge positions as an intermediate model between combinatorial and geometric link presentations. The explicit agreement between the global Dehornoy canonical braid and the cosetwise representative is a strength, as it leverages an established order without introducing circularity or new ad-hoc invariants beyond the fixed complexity function.

minor comments (3)
  1. [Abstract and §1] The abstract and introduction would benefit from a brief explicit statement of what constitutes a 'proper' complexity function (e.g., its key properties that guarantee the induced order on double cosets is strict and total), even if the full definition appears later.
  2. [Throughout] Notation for the double coset P_{2n}=H_{2n}∖B_{2n}/H_{2n} uses backslashes; consider using the standard LaTeX quotient notation or adding a clarifying sentence on the left/right actions of H_{2n} for readers less familiar with Hilden groups.
  3. [Final section] In the application to the bridge finiteness conjecture, include a one-sentence recall of the original conjecture statement to make the reformulation self-contained.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, for highlighting its significance in extending the Dehornoy order to the space of plat presentation classes, and for the recommendation of minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper's central construction fixes an external proper complexity function on B_{2n} and then applies the independently established Dehornoy order on braids to induce a total order on the double-coset space P_{2n}. The distinguished class CanPlat_D^{(n)}(L) and the agreement between the global Dehornoy canonical braid and the cosetwise representative are derived results from these definitions rather than inputs. No load-bearing step reduces by construction to a self-definition, a fitted parameter renamed as prediction, or a self-citation chain; the reformulation of the bridge-finiteness conjecture follows directly from boundedness of the resulting canonical representatives. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central construction relies on the prior Dehornoy ordering and standard facts about braid groups and Hilden groups; the complexity function is an additional choice fixed per n.

free parameters (1)
  • proper complexity function on B_{2n}
    Fixed for each n to allow the Dehornoy order to define the total order on the double cosets.
axioms (2)
  • standard math The Dehornoy order is a strict total order on the braid group B_m.
    This is a known property used to induce the order on the quotient set P_{2n}.
  • domain assumption H_{2n} is the Hilden group acting on the braid group.
    Used to form the double cosets representing plat presentations.

pith-pipeline@v0.9.0 · 5491 in / 1663 out tokens · 65329 ms · 2026-05-10T18:07:19.737415+00:00 · methodology

discussion (0)

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Reference graph

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