arxiv: 2605.01026 · v1 · submitted 2026-05-01 · 🧮 math.GT · math.QA

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A HOMFLYPT-type invariant for pseudo links via a resolution in Hecke algebras

Ioannis Diamantis

Pith reviewed 2026-05-09 18:01 UTC · model grok-4.3

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keywords pseudo linksHOMFLYPT polynomialHecke algebraskein relationsresolution mapOcneanu tracestate sum

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The pith

A resolution homomorphism produces a HOMFLYPT-type invariant for pseudo links.

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

The paper develops a way to assign a polynomial to pseudo links, which are link diagrams where some crossings have no specified over or under strand. It uses a map from the algebra of pseudo braids to the standard Hecke algebra that replaces each ambiguous crossing with a combination of the two possible classical crossings. This allows the existing Ocneanu trace on the Hecke algebra to be transferred to the pseudo setting, after suitable normalization, producing an invariant that obeys a skein relation adapted to pre-crossings. The construction also gives an explicit way to compute the invariant by summing the classical HOMFLYPT polynomials of all possible ways to resolve the pre-crossings into actual crossings.

Core claim

We construct a HOMFLYPT-type invariant for oriented pseudo links via the pseudo Hecke algebra of type A. The construction is based on a resolution homomorphism that maps each pseudo generator to a linear combination of a braid generator and its inverse, interpreting pre-crossings as algebraic superpositions of classical crossings. Composing this map with the Ocneanu trace and applying a suitable normalization yields an invariant satisfying a natural pseudo skein relation. We further show that the invariant admits a state-sum formulation as a weighted sum of classical HOMFLYPT-type invariants over all classical resolutions of the pseudo crossings, as well as a skein-theoretic characterization

What carries the argument

The resolution homomorphism from the pseudo Hecke algebra to the ordinary Hecke algebra, which sends each pseudo crossing generator to a linear combination of the corresponding braid generator and its inverse.

If this is right

The invariant satisfies a pseudo skein relation relating its value on a pre-crossing to its values on the two classical crossings.
It equals a weighted sum of classical HOMFLYPT invariants, one for each possible resolution of all pre-crossings in the diagram.
It is uniquely determined by its values on ordinary links together with the pseudo skein relation.
The invariant is unchanged under the pseudo Reidemeister moves when properly normalized.

Where Pith is reading between the lines

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

This state-sum view suggests that the invariant for a pseudo link can be calculated by enumerating all classical link resolutions and averaging their HOMFLYPT values with appropriate weights.
The skein characterization may allow recursive computation of the invariant directly on pseudo diagrams without enumerating resolutions.

Load-bearing premise

The proposed map from the pseudo Hecke algebra to the ordinary one must preserve all the algebraic relations and be compatible with the moves that define equivalence of pseudo links.

What would settle it

Compute the invariant for a specific pseudo link diagram and check whether it remains the same after performing a pseudo Reidemeister move that changes the diagram; if the value changes, the construction fails to produce an invariant.

Figures

Figures reproduced from arXiv: 2605.01026 by Ioannis Diamantis.

**Figure 1.** Figure 1: An example of a pseudo knot diagram containing one pre-crossing. view at source ↗

**Figure 2.** Figure 2: Reidemeister Moves for Pseudo Knots. Remark 2.2. Pseudo knots are closely related to singular knots, that is, knots with finitely many transverse double points regarded as rigid vertices. Indeed, there is a natural correspondence between singular diagrams and pseudo diagrams obtained by replacing each singular crossing with a pre-crossing [3, 6]. However, this correspondence is not an equivalence of theo… view at source ↗

**Figure 3.** Figure 3: The generators of PMn: classical crossings σ ±1 i and pre-crossings pi . Remark 2.4. The relations above are formally identical to the defining relations of the singular braid monoid after replacing each singular generator τi by the pseudo generator pi . Consequently, there is an isomorphism PMn ∼= SMn, σ±1 i 7→ σ ±1 i , pi 7→ τi , where SMn denotes the singular braid monoid [3]. This isomorphism is algeb… view at source ↗

**Figure 4.** Figure 4: A pseudo braid and its closure. 1. conjugation α ∼ β −1αβ, where α ∈ PMn and β ∈ Bn; 2. commuting αβ ∼ βα, for α, β ∈ PMn; 3. classical stabilization and destabilization α ∼ ασ±1 n ; 4. pseudo-stabilization α ∼ αpn. Remark 2.8. A sharpened version of the Markov theorem for pseudo links, formulated in terms of L-moves, was introduced in [5]. This approach parallels the classical L-move framework [15] and pr… view at source ↗

**Figure 5.** Figure 5: Conjugation, Stabilzation, Commuting and Pseudo Stabilization moves. view at source ↗

**Figure 6.** Figure 6: From left to right: a singular crossing L×, a positive crossing L+, a negative crossing L−, and the smoothing L0. Remark 2.12. Although the singular braid monoid SMn is algebraically isomorphic to the pseudo braid monoid PMn via the correspondence τi ↔ pi , the two theories differ at the topological level, due to the PR1 move. As a consequence, standard trace-based constructions do not directly yield invar… view at source ↗

read the original abstract

Pseudo links generalize classical links by allowing crossings with missing over/under information, called pre-crossings. While the pseudo braid framework provides an algebraic description of pseudo links via a Markov-type theorem, the construction of polynomial invariants using Hecke algebra techniques is obstructed by the presence of the pseudo Reidemeister 1 move. In this paper, we construct a HOMFLYPT-type invariant for oriented pseudo links via the pseudo Hecke algebra of type \(A\). The construction is based on a resolution homomorphism that maps each pseudo generator to a linear combination of a braid generator and its inverse, interpreting pre-crossings as algebraic superpositions of classical crossings. Composing this map with the Ocneanu trace and applying a suitable normalization yields an invariant satisfying a natural pseudo skein relation. We further show that the invariant admits a state-sum formulation as a weighted sum of classical HOMFLYPT-type invariants over all classical resolutions of the pseudo crossings, as well as a skein-theoretic characterization in terms of its values on classical links and the pseudo skein relation.

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 gives a clean algebraic resolution for a HOMFLYPT-type invariant on pseudo links that sidesteps the usual R1 obstruction and comes with a usable state-sum formula.

read the letter

The paper defines a resolution homomorphism from the pseudo Hecke algebra to the ordinary one, sending each pre-crossing generator to a linear combination of a braid generator and its inverse. Composing with the Ocneanu trace and normalizing produces an invariant that satisfies a natural pseudo skein relation. They also show it equals a weighted sum of classical HOMFLYPT values over all ways to resolve the pre-crossings into ordinary crossings. This directly addresses the pseudo Reidemeister 1 problem that has stopped earlier Hecke-algebra attempts on this class of objects. The state-sum version is the most practical part because it reduces computation to known classical invariants and supplies an independent check on the skein relation. The construction sits on top of the existing Markov theorem for pseudo braids and the standard trace, so there is no circularity or invented parameters. The abstract and stress-test description line up without internal contradictions. The main thing that still needs verification is whether the resolution map really preserves all the pseudo Hecke relations and commutes with the moves; if the full paper spells that out with explicit checks, the rest follows standard lines. This is niche work aimed at people who already use Hecke algebras for knot invariants and want to extend them to links with incomplete crossing data. A reader who knows the classical case will pick it up quickly and get a concrete new tool. It is worth sending to peer review because the resolution technique and the state-sum are new and checkable, even if the overall impact stays inside pseudo-link theory.

Referee Report

2 major / 3 minor

Summary. The paper constructs a HOMFLYPT-type invariant for oriented pseudo links by introducing a pseudo Hecke algebra of type A and defining a resolution homomorphism that sends each pseudo generator to a linear combination of a classical braid generator and its inverse. Composing this map with the Ocneanu trace on the Hecke algebra, followed by suitable normalization, produces an invariant that satisfies a natural pseudo skein relation. The construction is shown to be equivalent to a state-sum formula expressing the invariant as a weighted sum of classical HOMFLYPT-type invariants over all possible classical resolutions of the pseudo crossings, and it admits a skein-theoretic characterization based on its values on classical links together with the pseudo skein relation.

Significance. If the resolution homomorphism is a well-defined algebra homomorphism compatible with the relations of the pseudo Hecke algebra and the pseudo Reidemeister moves (including the pseudo R1 move that obstructed prior attempts), the result supplies a systematic algebraic extension of the classical HOMFLYPT polynomial to the setting of pseudo links. The state-sum formulation provides an independent computational verification route and a direct link to existing invariants, which strengthens the claim and may facilitate applications in distinguishing pseudo links or studying their properties.

major comments (2)

[Definition of the resolution homomorphism] The central claim rests on the resolution homomorphism being a well-defined algebra homomorphism from the pseudo Hecke algebra to the ordinary Hecke algebra. Explicit verification is required that this map preserves all defining relations of the pseudo Hecke algebra, especially those mixing pseudo generators with classical braid generators and inverses.
[Proof of invariance under pseudo Reidemeister moves] Invariance under the pseudo Reidemeister 1 move must be checked after normalization; the abstract notes this move as the prior obstruction, so the choice of normalization constants (likely involving the parameters of the linear combination) needs to be shown to cancel the contribution of this move when the Ocneanu trace is applied.

minor comments (3)

[Notation and definitions] Clarify the precise linear combination used for the resolution map (coefficients in terms of the Hecke algebra parameter) and confirm that it is independent of the choice of representative in the pseudo braid group.
[Examples and computations] Add a short table or explicit example computing the invariant on a simple pseudo link (e.g., a single pseudo crossing) to illustrate the state-sum formula and its reduction to the classical HOMFLYPT polynomial when all crossings are resolved.
[References] Ensure all references to the Markov-type theorem for pseudo braids and the standard Ocneanu trace are cited with precise statements of the external results being invoked.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive suggestions. We address the two major comments below and will incorporate the requested clarifications into the revised manuscript.

read point-by-point responses

Referee: [Definition of the resolution homomorphism] The central claim rests on the resolution homomorphism being a well-defined algebra homomorphism from the pseudo Hecke algebra to the ordinary Hecke algebra. Explicit verification is required that this map preserves all defining relations of the pseudo Hecke algebra, especially those mixing pseudo generators with classical braid generators and inverses.

Authors: We agree that a fully explicit verification is necessary for rigor. In the revised manuscript we will add a dedicated subsection (in Section 3) that checks preservation of every defining relation of the pseudo Hecke algebra, with particular attention to the mixed relations involving both pseudo generators and classical generators/inverses. The verification proceeds by direct substitution of the linear combination into each relation and use of the quadratic and braid relations already satisfied by the classical Hecke algebra. revision: yes
Referee: [Proof of invariance under pseudo Reidemeister moves] Invariance under the pseudo Reidemeister 1 move must be checked after normalization; the abstract notes this move as the prior obstruction, so the choice of normalization constants (likely involving the parameters of the linear combination) needs to be shown to cancel the contribution of this move when the Ocneanu trace is applied.

Authors: We accept the observation. The normalization constants are chosen exactly so that the pseudo R1 contribution vanishes. In the revision we will expand the invariance proof (currently in Section 4) to include an explicit computation: after applying the resolution map and the Ocneanu trace, the difference between the two sides of the pseudo R1 move is shown to be zero by direct cancellation using the specific coefficients in the linear combination and the known properties of the trace on the classical Hecke algebra. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper introduces a new resolution homomorphism from the pseudo Hecke algebra to the ordinary Hecke algebra that maps each pseudo generator to a linear combination of a braid generator and its inverse. This is composed with the standard Ocneanu trace and a normalization to produce the invariant. The Markov-type theorem for pseudo braids and the Ocneanu trace are external prior results; the central construction, pseudo skein relation, and state-sum formulation over classical resolutions do not reduce to fitted inputs, self-definitions, or unverified self-citations. The derivation chain remains independent of the final invariant values.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the existence of a pseudo braid group with a Markov-type theorem, the standard relations of the Hecke algebra of type A, and the assumption that the proposed linear combination for each pseudo generator extends to a homomorphism compatible with all relations. No new free parameters beyond the usual variables of the HOMFLYPT polynomial are introduced in the abstract.

axioms (2)

domain assumption The pseudo braid group admits a Markov-type theorem allowing invariants to be defined via traces on closures of pseudo braids.
Invoked to guarantee that the algebraic construction yields a link invariant.
ad hoc to paper The resolution map sending each pseudo generator to a linear combination of a braid generator and its inverse respects the defining relations of the pseudo Hecke algebra.
This is the load-bearing step that makes the homomorphism well-defined and allows the trace to produce an invariant.

pith-pipeline@v0.9.0 · 5476 in / 1777 out tokens · 27127 ms · 2026-05-09T18:01:04.486443+00:00 · methodology

discussion (0)

Reference graph

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