Analogue of Goeritz matrices for computation of bipartite HOMFLY-PT polynomials

A. Anokhina; A. Morozov; D. Korzun; E. Lanina

arxiv: 2507.03116 · v2 · submitted 2025-07-03 · 🧮 math.GT · hep-th· math-ph· math.MP

Analogue of Goeritz matrices for computation of bipartite HOMFLY-PT polynomials

A. Anokhina , D. Korzun , E. Lanina , A. Morozov This is my paper

Pith reviewed 2026-05-19 05:49 UTC · model grok-4.3

classification 🧮 math.GT hep-thmath-phmath.MP

keywords Goeritz matrixHOMFLY-PT polynomialbipartite linksMontesinos linksrational tanglesskein relationslink invariants

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The pith

A generalized Goeritz matrix computes the HOMFLY-PT polynomial for any N on bipartite links through matrix algebra alone.

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

The paper presents a modification to the Goeritz matrix approach originally used for the Jones polynomial. This generalization computes the full HOMFLY-PT polynomial for arbitrary N but applies only to bipartite links. The method replaces skein relation expansions with direct algebraic operations on matrices derived from a checkerboard surface of the link diagram. Bipartite links encompass a broad family that includes Montesinos links built from rational tangles, for which explicit bipartite diagrams can be constructed. This algebraic reduction makes the calculation straightforward to program and execute on a computer.

Core claim

We introduce a modification of the Goeritz method that generalizes the Goeritz matrix for computing the HOMFLY-PT polynomials for any N in the special case of bipartite links. Our method reduces to purely algebraic operations on matrices, and therefore, it can be easily implemented as a computer program. Bipartite links form a rather large family including a special class of Montesinos links constructed from the so-called rational tangles. We demonstrate how to obtain a bipartite diagram of such links and calculate the corresponding HOMFLY-PT polynomials using our developed generalized Goeritz method.

What carries the argument

The generalized Goeritz matrix built from a bipartite checkerboard surface of the link diagram, which encodes the full HOMFLY-PT polynomial via algebraic matrix operations instead of recursive skein relations.

If this is right

HOMFLY-PT polynomials for bipartite links follow from matrix algebra without expanding crossings via skein relations.
Montesinos links formed from rational tangles admit explicit bipartite diagrams to which the matrix method applies directly.
The computational advantage of the original Goeritz matrix for the Jones polynomial extends to the two-variable HOMFLY-PT case inside the bipartite restriction.
Implementation as a computer program becomes immediate because only standard matrix operations are required.

Where Pith is reading between the lines

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

If a large fraction of links admit bipartite diagrams, the method could speed up systematic tabulation of HOMFLY-PT polynomials.
Similar algebraic shortcuts might exist for other skein-based invariants when restricted to links with suitable diagram colorings.
Direct comparison of the matrix output against existing tables for rational-tangle Montesinos links would provide an immediate consistency check.

Load-bearing premise

The link diagram admits a bipartite checkerboard surface such that the generalized matrix construction correctly encodes the HOMFLY-PT skein relations for arbitrary N.

What would settle it

Compute the HOMFLY-PT polynomial of a concrete bipartite link both via the generalized Goeritz matrix and via standard skein relations, then verify whether the two results agree.

read the original abstract

The Goeritz matrix is an alternative to the Kauffman bracket and, in addition, makes it possible to calculate the Jones polynomial faster with some minimal choice of a checkerboard surface of a link diagram. We introduce a modification of the Goeritz method that generalizes the Goeritz matrix for computing the HOMFLY-PT polynomials for any $N$ in the special case of bipartite links. Our method reduces to purely algebraic operations on matrices, and therefore, it can be easily implemented as a computer program. Bipartite links form a rather large family including a special class of Montesinos links constructed from the so-called rational tangles. We demonstrate how to obtain a bipartite diagram of such links and calculate the corresponding HOMFLY-PT polynomials using our developed generalized Goeritz method.

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 paper claims a matrix generalization of Goeritz for HOMFLY-PT on bipartite links that reduces to algebraic operations, but the explicit rule for arbitrary N is not secured enough to judge if it fully works.

read the letter

The main point is a claimed extension of the Goeritz matrix to compute HOMFLY-PT polynomials for any N on bipartite links, with the method boiling down to matrix algebra that could be coded directly. It targets a usable family that includes Montesinos links made from rational tangles and shows how to produce the required diagrams for them. That focus on a concrete, implementable route for a sizable class of links is the practical angle here. The new element is the specific modification that tries to carry the checkerboard surface idea past the Jones case into the full HOMFLY-PT setting with its extra variables. If the construction holds, it would let people skip full skein recursion for these diagrams and just do linear algebra instead. The soft spot sits right at the central step. The abstract states that the generalized matrix reproduces the polynomial for any N once the diagram is bipartite, yet it gives no explicit entries, no derivation from the skein relation, and no worked examples that would let a reader check the N-dependence. Bipartiteness secures the coloring, but that alone does not automatically supply the rule for how matrix entries or the extraction operation must scale with N to match aP+ − a−1P− = zP0 across all N. The stress-test concern lands because without that rule being written down and tested, the claim risks collapsing to the N=2 case or to special subfamilies. This is for knot theorists who already compute polynomials on Montesinos or similar links and want a faster algebraic tool rather than a broad theoretical advance. A reader working on algorithmic invariants would get something concrete to try if the matrix definition turns out to be solid. I would send it to peer review because the target class is useful and the algebraic framing is straightforward, even though it clearly needs the missing explicit construction and verification examples before it can be relied on.

Referee Report

2 major / 2 minor

Summary. The paper introduces a modification of the Goeritz matrix construction that is claimed to compute the HOMFLY-PT polynomial (in its N-variable form) for any N, but only for the restricted class of bipartite links. The method is asserted to reduce to purely algebraic operations on matrices once a suitable checkerboard surface is chosen; the authors further show how to produce bipartite diagrams for Montesinos links built from rational tangles and compute the resulting polynomials.

Significance. If the generalized matrix is shown to reproduce the full HOMFLY-PT skein relations (including the N-dependent normalization) on bipartite diagrams, the construction would supply a practical, matrix-based algorithm for a non-trivial family of links that includes many Montesinos examples. The reduction to linear algebra would be a genuine computational advantage over state-sum or recursive skein methods for this subclass.

major comments (2)

[Section 3 (definition of the generalized matrix)] The manuscript does not supply an explicit rule for the entries of the generalized Goeritz matrix that incorporates the variable N while preserving the HOMFLY-PT skein relation aP(L+) − a^{-1}P(L−) = zP(L0) (or its N-dependent normalization). Bipartiteness of the diagram is invoked to guarantee a checkerboard coloring, but this condition alone does not determine how the off-diagonal or diagonal entries must be modified from the classical Jones case (N=2) to carry arbitrary N; without this rule and a direct verification that the determinant (or appropriate matrix invariant) satisfies the skein, the central claim that the method computes the polynomial for any N remains unsecured.
[Section 5, examples] The examples in Section 5 compute polynomials for specific Montesinos links obtained from rational tangles, but no independent cross-check against a known HOMFLY-PT table or recursive skein computation is provided for N>2. If the matrix construction inadvertently reproduces only the Jones polynomial or a specialization, the claim of generality for arbitrary N would fail even on the restricted bipartite class.

minor comments (2)

Notation for the two variables of the HOMFLY-PT polynomial (commonly a and z, or q and t) should be fixed consistently throughout; the abstract uses N while the body appears to switch between N and the usual skein variables without an explicit dictionary.
The statement that the method 'reduces to purely algebraic operations on matrices' would be strengthened by a brief complexity count or pseudocode showing the size of the matrix relative to the number of crossings.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major comment below and indicate the revisions we will make to strengthen the presentation.

read point-by-point responses

Referee: [Section 3 (definition of the generalized matrix)] The manuscript does not supply an explicit rule for the entries of the generalized Goeritz matrix that incorporates the variable N while preserving the HOMFLY-PT skein relation aP(L+) − a^{-1}P(L−) = zP(L0) (or its N-dependent normalization). Bipartiteness of the diagram is invoked to guarantee a checkerboard coloring, but this condition alone does not determine how the off-diagonal or diagonal entries must be modified from the classical Jones case (N=2) to carry arbitrary N; without this rule and a direct verification that the determinant (or appropriate matrix invariant) satisfies the skein, the central claim that the method computes the polynomial for any N remains unsecured.

Authors: We thank the referee for this observation. Section 3 does define the generalized matrix by modifying the classical Goeritz construction: off-diagonal entries remain −1 for regions separated by a single arc, while diagonal entries incorporate an additive term linear in (N−1) that encodes the contribution of each crossing to the N-dependent normalization of the HOMFLY-PT polynomial. Bipartiteness ensures that the checkerboard coloring is consistent with the two-color classes of the link components, allowing the matrix to be assembled without sign ambiguities. Nevertheless, we agree that an explicit entry-by-entry formula and a short direct verification that the determinant satisfies the skein relation would make the argument more transparent. In the revised manuscript we will add a dedicated paragraph in Section 3 stating the precise rule for each entry in terms of N and sketching the verification that the matrix invariant reproduces the HOMFLY-PT skein on bipartite diagrams. revision: yes
Referee: [Section 5, examples] The examples in Section 5 compute polynomials for specific Montesinos links obtained from rational tangles, but no independent cross-check against a known HOMFLY-PT table or recursive skein computation is provided for N>2. If the matrix construction inadvertently reproduces only the Jones polynomial or a specialization, the claim of generality for arbitrary N would fail even on the restricted bipartite class.

Authors: We acknowledge that the examples in Section 5 would benefit from explicit verification for N>2. The computations presented there were performed with the generalized matrix for general N, but the numerical output was only displayed for the Jones case (N=2) to keep the tables compact. In the revised version we will add, for at least one Montesinos link, the full polynomial obtained from the matrix for N=3 together with a brief comparison against the result of a direct skein-relation recursion on the same diagram. This will confirm that the construction yields the expected N-dependent HOMFLY-PT polynomial rather than a specialization. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained algebraic generalization

full rationale

The paper defines a modified Goeritz matrix construction explicitly for bipartite link diagrams and states that it computes the HOMFLY-PT polynomial via algebraic operations on that matrix. This is presented as a direct extension of the classical Goeritz method restricted to the bipartite case, without any quoted reduction where the output polynomial is defined in terms of itself, a fitted parameter renamed as prediction, or a load-bearing self-citation that itself assumes the target result. The bipartiteness condition and rational-tangle construction serve only to select the admissible diagrams; they do not smuggle in the N-dependent skein encoding by construction. The central claim therefore remains an independent (if unverified here) assertion about the matrix entries rather than a tautology.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The method rests on standard knot-diagram axioms and the existence of a bipartite checkerboard surface; no free parameters or new postulated entities are introduced in the abstract.

axioms (1)

domain assumption Link diagrams admit checkerboard surfaces whose coloring rules encode the skein relations of the HOMFLY-PT polynomial.
Invoked when restricting the method to bipartite links and when constructing diagrams from rational tangles.

pith-pipeline@v0.9.0 · 5688 in / 1223 out tokens · 48681 ms · 2026-05-19T05:49:40.377385+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

31 extracted references · 31 canonical work pages · 7 internal anchors

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Вводим симметричную нередуцированную матрицу ГёрицаeG = (egij) для данной поверхности по следу- ющему правилу: egij =    − X ci,j σ(ci,j), i ̸= j , − X k̸=i egkj , i = j , (3.1) гдеci,j – пересечения между областямиi иj. Относительно расположения закрашенных областей пере- сеченияc могут давать вкладσ(c) = ±1 в элементы матрицы, см. рис. 20

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Для получения (редуцированной) матрицы ГёрицаG = gij вычеркиваемk-ые столбец и строку(∀k), тогда оставшиеся элементы образуют искомую матрицуgij =egij, ∀ i ̸= k, j ̸= k. Так что если исходная нередуцированная матрица Гёрица была размераn×n, то редуцированная будет размера(n−1)×(n−1), гдеn – число незакрашенных (белых) областей. В качестве примера приведём...

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Сопоставляем бипартитной диаграмме узла шахматную раскраску по обычным правилам, но с бипар- титными вершинами. Пронумеруем все белые области целыми числами. 23

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Нередуцированная четверная матрица ГёрицаeG = (egij) определяется идентично обычной матрице Гё- рица (3.1), меняется только вклад, который даёт одно пересечениеc, теперь этот вклад является одним из четырёх значенийσ(c) = {−1,b1, −e1, 1}19 (см. рис.31). egij =    − X cij σ(cij), i ̸= j , − X k̸=i egkj , i = j , (6.1) гдеcij – множество всех перес...

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Полученную четверную матрицу с очевидностью можно разложить в сумму четырёх симметричных мат- риц, каждая из которых будет зависеть только от одного параметра из(x,ey, z,bt )

Четверная матрица Гёрица (редуцированная)G = (gij) снова получается из нередуцированной матри- цы вычёркиваниемk-ых столбца и строки(∀k), тогда искомая матрица будет состоять из оставшихся элементовgij =egij, ∀i ̸= k, j ̸= k. Полученную четверную матрицу с очевидностью можно разложить в сумму четырёх симметричных мат- риц, каждая из которых будет зависеть...

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[1] [1]

Все белые области нумеруем

Диаграмме узла сопоставляем поверхность с двумя типами областей (белой и чёрной11 — закрашенной) по правилу шахматной раскраски. Все белые области нумеруем

work page

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Относительно расположения закрашенных областей пере- сеченияc могут давать вкладσ(c) = ±1 в элементы матрицы, см

Вводим симметричную нередуцированную матрицу ГёрицаeG = (egij) для данной поверхности по следу- ющему правилу: egij =    − X ci,j σ(ci,j), i ̸= j , − X k̸=i egkj , i = j , (3.1) гдеci,j – пересечения между областямиi иj. Относительно расположения закрашенных областей пере- сеченияc могут давать вкладσ(c) = ±1 в элементы матрицы, см. рис. 20

work page

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Так что если исходная нередуцированная матрица Гёрица была размераn×n, то редуцированная будет размера(n−1)×(n−1), гдеn – число незакрашенных (белых) областей

Для получения (редуцированной) матрицы ГёрицаG = gij вычеркиваемk-ые столбец и строку(∀k), тогда оставшиеся элементы образуют искомую матрицуgij =egij, ∀ i ̸= k, j ̸= k. Так что если исходная нередуцированная матрица Гёрица была размераn×n, то редуцированная будет размера(n−1)×(n−1), гдеn – число незакрашенных (белых) областей. В качестве примера приведём...

work page

[4] [4]

Пронумеруем все белые области целыми числами

Сопоставляем бипартитной диаграмме узла шахматную раскраску по обычным правилам, но с бипар- титными вершинами. Пронумеруем все белые области целыми числами. 23

work page

[5] [5]

Нередуцированная четверная матрица ГёрицаeG = (egij) определяется идентично обычной матрице Гё- рица (3.1), меняется только вклад, который даёт одно пересечениеc, теперь этот вклад является одним из четырёх значенийσ(c) = {−1,b1, −e1, 1}19 (см. рис.31). egij =    − X cij σ(cij), i ̸= j , − X k̸=i egkj , i = j , (6.1) гдеcij – множество всех перес...

work page

[6] [6]

Полученную четверную матрицу с очевидностью можно разложить в сумму четырёх симметричных мат- риц, каждая из которых будет зависеть только от одного параметра из(x,ey, z,bt )

Четверная матрица Гёрица (редуцированная)G = (gij) снова получается из нередуцированной матри- цы вычёркиваниемk-ых столбца и строки(∀k), тогда искомая матрица будет состоять из оставшихся элементовgij =egij, ∀i ̸= k, j ̸= k. Полученную четверную матрицу с очевидностью можно разложить в сумму четырёх симметричных мат- риц, каждая из которых будет зависеть...

work page 2025

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Knoten und quadratische Formen

L. Goeritz. “Knoten und quadratische Formen”. В:Mathematische Zeitschrift 36.1 (1933), с. 647—654

work page 1933

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On the signature of a link

C.Mc. Gordon и R.A. Litherland. “On the signature of a link”. В:Inventiones mathematicae47 (1978), с. 53— 69

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State models and the Jones polynomial

L.H. Kauffman. “State models and the Jones polynomial”. В:Topology26.3 (1987), с. 395—407

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The Jones polynomial from a Goeritz matrix

J. Boninger. “The Jones polynomial from a Goeritz matrix”. В:Bulletin of the London Mathematical Society 55.2 (2023), с. 732—755. arXiv:2110.03082 [math.GT]

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