Path to homology of Yang-Baxter operators

Jozef H. Przytycki

Path to homology of Yang-Baxter operators

T0 review · 0 major / 5 minor · reviewed 2026-07-31 · grok-4.5

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Pith's one-line read Yang-Baxter operators that yield classical link polynomials carry a computable homology built from the same face maps that encode the third Reidemeister move.

desk verdict Autobiographical path-survey with real new low-dim YB homology formulas and an open RIII face-map observation; solid but not a theorem paper. read the letter →

arxiv 2607.28626 v1 pith:IWAS3HOZ submitted 2026-07-30 math.GT math.HO

Jozef H. Przytycki This is my paper · ORCID

classification math.GTmath.HO MSC 57K1016T2557K3157-03

keywords knotdistributivehomologyquandleskeinmoduleYang-Baxteroperatorhistoryoftheory

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper traces a path from the early history of knot theory through nonassociative algebras, skein modules, n-moves, and distributive (quandle/rack) homology to a homology theory for general Yang-Baxter operators. The author shows how entropic and self-distributive conditions arise naturally when link calculations must be independent of the order of local moves, then lifts those conditions to a precubic chain complex whose face maps are read off curtain diagrams of the operator. The novel contributions are an explicit decomposition of the third Reidemeister move into a signed sum of cubic face maps and closed-form calculations of low-dimensional homology for the column-unital operators that recover the HOMFLYPT polynomial. A sympathetic reader cares because the construction gives a single algebraic home to both classical coloring invariants and the operators behind the Jones polynomial, and supplies concrete groups that can be used for cocycle invariants and for testing deeper links with other homological knot invariants.

What carries the argument

Precubic face maps d^ℓ_i and d^r_i of a Yang-Baxter operator R (visualized by curtain diagrams): their alternating difference is the boundary operator; column-unitality lets walls absorb colors so the precubic identities close and Reidemeister III becomes a homology relation.

What would settle it

For small m (e.g. m=2 or 3) build the chain groups of the column-unital HOMFLYPT operator R^(m) over Z[y] or Z[y^{±1}], compute H_2 and H_3 by linear algebra, and check whether free rank and torsion match the stated formulas.

Watch

Extended reading notes

Core claim

Column-unital Yang-Baxter operators, especially the family that produces the HOMFLYPT polynomial, admit a well-defined precubic homology theory; their second homology is given explicitly by a free part plus torsion of orders dividing 1-y^2 and 1-y^4, and the third Reidemeister move itself decomposes as the boundary of a cubic chain built from the same left and right face maps.

Load-bearing premise

Each column of the Yang-Baxter matrix must sum to one; without that absorption property the boundary is not defined the same way and the Reidemeister-III argument does not go through.

Editorial extensions

If this is right

Low-dimensional homology of HOMFLYPT Yang-Baxter operators becomes available as a source of cocycle invariants of links.
The Reidemeister-III face-map decomposition supplies a diagrammatic certificate that the homology is unchanged by that move in the unital case.
Higher homology groups of these operators are computable in principle and are already partially known through dimension six.
The same face-map language unifies set-theoretic quandle/rack homology with operator-level Yang-Baxter homology.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The curtain model and Reidemeister decomposition suggest a comparison map from Yang-Baxter homology toward Khovanov homology that the paper only dreams of and does not construct.
Torsion patterns already visible in the HOMFLYPT operators may eventually encode Burnside-type finiteness phenomena for links, linking the earlier n-move sections to the homology calculations.
Dropping column-unitality would require wall correction terms and could recover a larger class of statistical-mechanical models at the price of a more complicated complex.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The punchline is that this is mostly Przytycki telling the story of how he got from Conway algebras and n-moves through distributive homology to Yang-Baxter homology, with two concrete new pieces: the explicit H2 calculation for the column-unital HOMFLYPT operators (Theorem 7.2) and the geometric decomposition of Reidemeister III into cubic face maps (Figure 27 / Section 6). The rest is careful historical survey plus re-exposition of his earlier work.

What works: the algebraic setup is clean. Presimplicial and precubic modules, face maps, and the boundary for column-unital R are standard and correctly written. The H2 formula is derived by direct expansion with recursive polynomials and a computational tree (Figure 28); the conjectures on higher homology are stated honestly and some have already been settled in follow-ups noted in the e-print. History is documented with primary sources. The column-unital condition is flagged explicitly, so the scope is clear.

Soft spots are proportional and minor. Large stretches recycle the author’s own papers; novelty sits in the middle. Figure 27 is presented as an observation that “awaits further exploration,” not a finished invariance theorem, so it does not over-claim. Higher homology remains computational and open in places. No circularity or hidden gaps in the math itself.

This is for people already in quandle/YB homology or the history of modern knot invariants. A reading-group session on Sections 5–7 would be useful; the early history is optional. It deserves a serious referee rather than a desk reject—sound, reproducible pure-math exposition with a few original calculations and an intriguing open geometric remark. I would engage with the new homology formulas and the RIII picture; the rest is background I already know.

Referee Report

0 major / 5 minor

Summary. This extended version of two 2023 workshop talks combines a historical survey of knot theory (from Leibniz and Euler through Gauss, Listing, and modern polynomials) with an autobiographical account of the path from Conway algebras, entropic magmas, n-moves, Burnside groups of links, and quandle/distributive homology to a precubic homology theory for Yang-Baxter operators. The novel material centers on the construction of homology for column-unital Yang-Baxter operators via graphical face maps (Section 5–7), the observation that the third Reidemeister move decomposes into a signed sum of cubic face maps (Section 6, Figure 27), and explicit low-dimensional computations for the family of column-unital operators R^{(m)} that yield the HOMFLYPT polynomial (Theorem 7.1–7.2 and Conjectures 7.3–7.5), including the formula H_2(R^{(m)}) = k^{1+\binom{m}{2}} \oplus (k/(1-y^2))^{\binom{m}{2}} \oplus (k/(1-y^4))^{m-1}.

Significance. The paper supplies a coherent, well-documented narrative linking classical knot-theoretic constructions (skein modules, n-moves, Burnside groups, quandle homology) to a general precubic homology for column-unital Yang-Baxter operators. The explicit H_2 computation for the HOMFLYPT family (Theorem 7.2) and the recursive formulas for n-moves in higher-degree skein modules (Sections 3.6–3.7) are concrete, checkable contributions. The graphical face-map calculus and the RIII decomposition of Figure 27 open a natural line of inquiry connecting Yang-Baxter homology to Reidemeister invariance and, potentially, to Khovanov homology. As a survey-plus-research hybrid it is valuable for the Banach Center Proceedings audience and for researchers entering the area.

minor comments (5)

[§3.6–3.7] Several formulas and figures are dense (e.g., the recursive expansions of Dn in §3.6–3.7 and the computational tree of Figure 28). A short summary table of the closed formulas for U_n,k^{(m)} would improve readability.
[§6, Figure 27] Figure 27 is presented as an observation that 'awaits further exploration.' A one-sentence statement of precisely what is proved versus what remains conjectural would prevent over-reading.
[throughout] Typographical slips appear throughout (e.g., 'worksho p', 'historica l', 'infuenced', 'desribe', 'B¸ edlewo'). A careful copy-edit pass is needed.
[§7.3 and footnotes] The 'Added for e-print' notes (solved conjectures, new arXiv preprints) are useful but should be integrated cleanly into the main text or a short addendum so that the published version is self-contained.
[§4] Cross-references to the author’s earlier papers are frequent; a short 'notation and conventions' paragraph early in §4 would help readers who have not followed the whole series.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: survey-plus-computation paper whose homology formulas follow by direct expansion of stated precubic boundaries, not by tautology or load-bearing self-citation.

full rationale

The manuscript is an autobiographical survey culminating in two concrete algebraic claims: (i) a precubic homology for column-unital Yang–Baxter operators, with an explicit H2 formula for the HOMFLYPT family R(m) (Thm 7.2), and (ii) a geometric decomposition of Reidemeister III into signed cubic face maps (Fig. 27 / Sec. 6). Both rest on definitions given in the paper (precubic face maps dℓi, dri from the graphical model of Fig. 26; column-unitality so walls absorb; the explicit matrix entries of Thm 7.1). The H2 calculation is a finite expansion of ∂2 on those entries and is externally checkable; it is not forced by fitting or by renaming an input. Fig. 27 is offered as an observation that “still awaits further exploration,” not as a proved invariance theorem, so it does not over-claim. Heavy self-citation (Prz8, Prz10–11, PrWa2, etc.) is normal for a path-survey and is not load-bearing for the new formulas: those formulas do not reduce to the cited statements by construction. No self-definitional loop, no fitted-as-prediction step, no uniqueness theorem imported to forbid alternatives, and no ansatz smuggled in via citation. Score 1 reflects only the ordinary presence of author self-citation in a survey, not circular reasoning.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper works entirely inside standard algebraic topology and knot theory. No free parameters are fitted; the only numerical choices are the conventional normalizations of the HOMFLYPT Yang-Baxter matrices (column-unitalization). Background axioms are the usual ones of chain complexes, the Yang-Baxter equation, and Reidemeister moves. No new physical or mathematical entities are postulated beyond the homology theories already introduced in the author's earlier papers.

assumptions (4)

standard math A chain complex satisfies \partial^{2} = 0; homology is ker \partial / im \partial.
Used throughout Sections 4-7 to define distributive and Yang-Baxter homology.
domain assumption The Yang-Baxter equation (R \otimes id)(id \otimes R)(R \otimes id) = (id \otimes R)(R \otimes id)(id \otimes R) holds for the operators under consideration.
Invoked to guarantee that the face maps satisfy the precubic identities (Section 5).
domain assumption Column-unitality: each column of the matrix R sums to 1.
Required so that the wall-absorption condition holds and the boundary operator is well-defined without extra terms (Section 7).
domain assumption Reidemeister moves generate ambient isotopy of links in R^{3} (or S^{3}).
Background fact used to motivate invariance of state sums and homology under the third move (Sections 2 and 5-6).

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Pith. "Pith review of Path to homology of Yang-Baxter operators." pith.science (2026). https://pith.science/paper/IWAS3HOZ

@misc{pith2026260728626,
  author       = {Pith},
  title        = {Pith review of: Path to homology of Yang-Baxter operators},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IWAS3HOZ}},
  note         = {Machine review of arXiv:2607.28626}
}

read the original abstract

This paper is an extended version of two talks I gave during workshop ``Loops'13" in Bedlewo in June 2023. In the first talk I gave a historical introduction to Knot Theory. In the second, I traced my journey toward Yang-Baxter homology and this talk has a partially survey and a partially novel character.

Figures

Figures reproduced from arXiv: 2607.28626 by the authors.

**Figure 1.** Stamp seal, about 1700 BC (the British Museum) On the octagonal base [of hammer-handled haematite seal] are patterns surrounding a hieroglyphic inscription (largely erased). Four of the sides are blank and the other four are engraved with elaborate patterns typical of the period (and also popular in Syria) alternating with cult scenes...([Col], p.93) [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

**Figure 2.** Snake with Interlacing Coil. Cylinder seal. Ur, Mesopotamia. The Royal Cemetery, Early Dynastic period, c. 2600-2500 B.C. Lapis lazuli. Iraq Museum. Photograph courtesy of the British Museum, UI 9080, [Wo-Kr] [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** The crossed noose; “For the tying the crossed noose, a cord, folded double, is procured, and the ends of the cord are held in the left hand, and the loop is held in the right hand. Then the loop is twisted so that the slack parts of the cord crossed. Hence the noose is called crossed. After the slack parts of the cord have been crossed, the loop is placed on the crossing, and the lower slack part of the cord is pull… view at source ↗

Figures from the paper (24 more)

**Figure 4.** Figure 4: A knot by Leonardo [Mac]; c. 1496 We would argue, that modern knot theory has its roots with Gottfried Wilhelm Leibniz (1646-1716) speculation that aside from calculus and analytical geometry there should exist a “geometry of position” (geometria situs) which deals wit…

**Figure 5.** Figure 5: Six knots by D¨urer [Kur]; c. 1505-1507 in K¨onigsberg. He studied at the Pedagogicum there, and in 1733 settled in Danzig as a mathematics professor at the Academic Gymnasium (he was also a co-founder of the Nature Society and the first person to suggest the geometric…

**Figure 6.** Figure 6: Seven bridges of K¨onigsberg [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]

**Figure 7.** Figure 7: Knots of Vandermonde use in practice. The craftsman who fashions a braid, a net, or some knots will be concerned, not with questions of measurement, but with those of position: what he sees there is the manner in which the threads are interlaced. 2.2. Knot Theory from …

**Figure 8.** Figure 8: Meshing knot, 10’th knot of Gauss from 1794 a b c d 1 2 3 4 6 5 b c d 4 2 3 2 4 3+i 11a 2+i 1 4 3+i 3+i 1 4 2+2i 2+2i 1 4 3+2i 2+2i 1 3 4+3i Veraindrungder Coordiniz [PITH_FULL_IMAGE:figures/full_fig_p010_8.png]

**Figure 9.** Figure 9: It is a good method of coding a knotting (from a Gauss’ notebook (Handb.7)). Gauss coordinates are not always consistent; most of the time i is pointing downward but there are exceptions date the drawing; one can say for sure that it was done between 1814 and 1830, I w…

**Figure 10.** Figure 10: Framed tangle from Gauss’ notebook [Ga-1] two links are substantially different: and . Gauss’ analytical method has recently been revitalized by Witten’s approach to knot theory [Wit]. James Clerk Maxwell (1831-1879), in his fundamental book of 1873 “A treatise on ele…

**Figure 11.** Figure 11: The link of Maxwell paper on electrical circuits [Kir]. It has deep connections with knot theory, however the relations were discovered only about a hundred years later (e.g. the Kirchhoff complexity of a circuit corresponds to the determinant of the knot or link det…

**Figure 12.** Figure 12: Oriented skein triple for the skein (HOMFLYPT) polynomial Jones polynomial, skein polynomial (later called HOMFLYPT polynomial). I wanted to go there, I was already at PKS (bus) stop, but it was so crowded that I finally gave up travel, and I got cold as it was very …

**Figure 13.** Figure 13: Quartic relation 1 0 2 3 8 b +b +b +b +b = 0 4 +b [PITH_FULL_IMAGE:figures/full_fig_p018_13.png]

**Figure 15.** Figure 15: The smallest counterexample to Montesinos-Nakanishi 3- move conjecture, up to mirror image; (σ2σ3σ2σ −1 4 σ −1 1 ) 4 20-crossings (the Chen link) 3−move 3−move D0 D 3 3−move [PITH_FULL_IMAGE:figures/full_fig_p021_15.png]

**Figure 16.** Figure 16: 3-move and reduction of the trefoil knot and the figure eight knot to a trivial link. We found a counterexample by introducing the concept of Burnside groups of links and showing that the nth group is preserved by n moves; see [DaPr1, DaPr2, DaPr3]. Let us recall firs…

**Figure 17.** Figure 17: Core group relation at a crossing c = ba−1 b Already Burnside proved that for n = 3 the group B(r, 3) is finite [Bur]. Furthermore Levi and van der Waerden [L-W] proved that the group B(r, 3) has 3n+( n 2)+( n 3) elements; in particular, |B(3, 3)| = 37 and |B(4, 3)| =…

**Figure 18.** Figure 18: Core group relations applied to n-move; b goes to (ba−1 ) n b and a goes to (ba−1 ) na third Burnside group from third Burnside groups of all trivial links (|B3(Chen)| = 310). What was helping us at initial stages of our work was that Mike Newman put the presentation …

**Figure 19.** Figure 19: Quandle coloring on n-move; wn denotes the word of length n + 1 in which a and b alternate and the last letter is b, except that w0 = a. Note that w2 def = b ∗ (a ∗ b) Idem= (b ∗ b) ∗ (a ∗ b) distr = (b ∗ a) ∗ b 33I gave talk about our work at University of Maryland t…

**Figure 20.** Figure 20: Graphical interpretation of the axiom for X-shelf-set Y ; (y ∗ x1) ∗ x2 = (y ∗ x2) ∗ (x1 ∗ x2) 40C N n denotes the normalized chain complex, that is the quotient of Cn by the degenerate complex C D n . X X X i d i ( ) * Y X X X i [PITH_FULL_IMAGE:figures/full_fig_p03…

**Figure 21.** Figure 21: Graphical interpretation of a one term face map d (∗) i [PITH_FULL_IMAGE:figures/full_fig_p031_21.png]

**Figure 22.** Figure 22: Boltzmann weights R a,b c,d and R¯c,d a,b for positive and negative crossings We can now generalize the number of colorings to state sum (basic notion of statistical physics) by multiplying Boltzmann weight over all crossings and adding over all colorings: col(X;BW)(…

**Figure 23.** Figure 23: Invertibility of R and the parallel second Reidemeister move state sum is invariant under “parallel” (directly oriented) second Reidemeister move, see [PITH_FULL_IMAGE:figures/full_fig_p034_23.png]

**Figure 24.** Figure 24: Yang-Baxter equation from the positive third Reidemeister move For a given pre-Yang-Baxter operator we attempt to find presimplicial module, from which homology will be derived [PITH_FULL_IMAGE:figures/full_fig_p034_24.png]

**Figure 25.** Figure 25: Various possible interpretations of the graphical face map di ; one may find many more depending on needs; for 1-term Yang-Baxter homology, di drawn in the bottom right corner was one of the first to be considered shown in [PITH_FULL_IMAGE:figures/full_fig_p035_25.png]

**Figure 26.** Figure 26: Graphical interpretation of the face map di = d ℓ i − d r i Our graphical model allows quite straightforward calculation as it can be seen in the following example. Example 5.1. Assume R : X × X → X × X generates set-theoretic Yang-Baxter operator with R(x, y) = (R1(x…

**Figure 27.** Figure 27: Reidemeister third move and face maps d ε i crossing case. Furthermore, our column unital condition yields a relation R a,b a,b+R a,b b,a = 1 between entries of R, (see Subsection 7.2 for details). 7.1. Yang–Baxter homology from the quantum topology viewpoint. I wrote…

**Figure 28.** Figure 28: Computational tree for d ℓ 4 (a, b, c, d) where a ≤ b ≤ c ≤ d. We sum over all leaves with coefficients being the product of weights of edges from the root to the given leaf References [Adj] S. I. Adjan, The Burnside Problem and Identities in Groups (trans. J. Lennox …

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