REVIEW 4 major objections 5 minor 10 references
A simple algorithm to compute link polynomials defined by using skein relations
T0 review · 4 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash
Pith's one-line read By ordering all braid representatives, the paper reduces the HOMFLY skein recursion to a terminating induction and obtains a complete link invariant from the unique minimal representative.
desk verdict A real algorithmic idea for order-driven skein computation, but the termination proof is missing and the paper's finiteness justification is false. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing object is the total order $<_{\mathrm{br}}$ on braid representatives (Definition 2.2), which compares representatives first by word length, then by number of strands, then by the vector of weights $w_m$ counting how many generators of each absolute value appear, then by the absolute values of the entries, and finally by their signs. The algorithm also uses the ordered leading tag length $\mathrm{ol}(\beta)$, the length of the initial run of entries whose absolute values increase by one, to decide which of three stopping cases applies. The order is what makes the recursion well-founded: each allowed replacement lowers the representative, and the terminal forms are exactly the places where the skein move sends both branches to still-lower representatives.
What would settle it
Run Algorithm 3.4 on every braid representative with, say, at most five strands and twelve generators; if any input produces a cycle, or terminates at a braid $\gamma=[m;c_1,\ldots,c_l]$ with $\mathrm{ol}(\gamma)=q>0$ that is not covered by $c_{q+1}=c_q$ or by $c_{q+1}=c_q-1$ with $\operatorname{sgn}(c_{q+1})=-\operatorname{sgn}(c_q)$, then Lemma 3.5 and Theorem 3.6 fail.
Extended reading notes
Core claim
The central claim is Theorem 3.6: given a braid representative of a link $L$, the HOMFLY polynomial of $L$ can be computed inductively by the skein relation $\ell P(L_+)+\ell^{-1}P(L_-)+mP(L_0)=0$ together with the normalization $P(\mathrm{unknot})=1$, using Algorithm 3.4 as the reduction engine. At a terminal representative $\gamma$, the two skein branches are $\gamma'$, obtained by changing the sign of the $q$-th generator, and $\gamma''$, obtained by deleting it; the proof shows both are strictly smaller than $\gamma$ in the total order $<_{\mathrm{br}}$, and the empty braid represents the unknot. Because each representative has only finitely many smaller representatives, the induction is well-founded. The paper further claims that the unique minimal representative under this order is a complete link invariant.
Load-bearing premise
The argument rests on Algorithm 3.4 always terminating in one of the three listed terminal forms, with every nonterminal step strictly lowering the representative in the order; the paper's proof of this is a restatement of the cases, and Step 1 is a rotation rather than an order-decrease, so the decreasing measure is not fully exhibited.
Editorial extensions
If this is right
- For any braid input, HOMFLY computation becomes a deterministic induction: at each nontrivial terminal braid exactly one crossing is flipped or deleted, and both resulting representatives are smaller in $<_{\mathrm{br}}$.
- The algorithm is not specific to HOMFLY: the proof uses only the skein relation and the unknot normalization, so the same reduction computes any link polynomial defined by that skein relation.
- Every link gets a unique minimal braid representative under $<_{\mathrm{br}}$, so the order supplies a complete link invariant; whether that representative can be found efficiently is a separate question the paper does not settle.
- The recorded computation of HOMFLY polynomials for 2977 knots with braid crossing number at most 12 took 430 seconds in total, with an average node-growth factor of about $1.42^c$, suggesting the branching is much gentler than the generic $2^c$ of naive skein expansion.
Reading between the lines
- Because Step 1 of Algorithm 3.4 is a cyclic rotation rather than an order-decrease, a fully formal termination proof needs an invariant that treats rotations as neutral; searching for such a measure could either complete the proof or expose a hidden cycle.
- If Algorithm 3.4 can be shown to compute the minimal representative itself, the total order would turn the complete invariant into a practical classification tool for links.
- The measured growth factor of $1.42^c$ is an average over 2977 knots, not a worst-case bound; testing on longer braid words or on links with many components would show whether the practical improvement persists.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper proposes a total order <br on the set of all braid representatives and uses it to build an algorithm (Algorithm 3.4) that repeatedly replaces a braid representative by an equivalent one that is smaller in this order. The author claims that this algorithm terminates in one of three normal forms and that, in the non-unknot cases, applying the skein relation at a distinguished crossing reduces the computation of the HOMFLY polynomial to strictly smaller braid representatives. Theorem 3.6 states that the HOMFLY polynomial of any link can therefore be computed inductively. The paper also claims, as a by-product, that the minimal braid representative in each equivalence class is a complete link invariant. A sample computation for a six-crossing knot and a Mathematica experiment for knots up to 12 crossings are reported.
Significance. If the main theorem were fully proved, the paper would provide a concrete, order-driven recursion for skein-defined link polynomials and a potentially practical computational scheme; the reported average branching factor of 1.42 is suggestive. The paper is honest that the method is exponential in the worst case, consistent with known hardness results. However, the proposed complete invariant is definitional: choosing the minimum of an arbitrary total order within each equivalence class defines an invariant tautologically, and the paper does not address its computability or topological content. The central value of the paper therefore rests entirely on the termination and strict-descent claims for Algorithm 3.4.
major comments (4)
- [Section 3, Lemma 3.5] The proof of Lemma 3.5 does not prove termination; it only lists the three stopping rules and says that the three cases are exactly those terminated at Step 1, Case 4.2, and Case 4.3.1. The surrounding claim that every step decreases in <br is not literally true: Step 1 and parts of Step 4 are cyclic rotations, which are equivalence moves and can increase the representative in the order. For example, [2;1,2] rotates to [2;2,1], and [2;1,2] <br [2;2,1]. Since Theorem 3.6 relies on strict <br descent at every recursive branch, the well-foundedness of the algorithm's transition relation is the load-bearing premise, and it is neither proved nor checked case by case.
- [Section 2, before Theorem 2.3] The statement 'For any given braid representative β, there are finitely many braid representatives which are smaller than β' is false. For any nonempty representative β=[n;b1,...,bk] with k>0, every empty representative [r;-] with r>=1 is smaller by condition (1) of Definition 2.2, since 0<k. Hence there are infinitely many smaller representatives. The proof of Theorem 2.3 is therefore invalid as written; if the intended claim is well-foundedness of the restriction of <br to each equivalence class, that must be proved separately and is not supplied.
- [Lemma 3.5(3) and Theorem 3.6, case (3)] The terminal condition 'cq+1 = cq−1, and sgn(cq+1) = −sgn(cq)' is impossible as written: if cq>0 then cq−1 has positive sign, and if cq<0 then cq−1 has negative sign, so in both cases the signs are equal, not opposite. The intended condition, matching Subcase 4.3.1, appears to be |cq+1|=|cq|−1 with opposite signs. As printed, the subsequent verification that γ' is smaller than γ in Theorem 3.6 cannot be checked, and the displayed equivalence to δ in that proof needs to be re-verified after correcting the condition.
- [Theorem 3.6, final paragraph] In case (3) of Lemma 3.5, the proof asserts that γ' is equivalent to δ by elementary relations (5) and (2), citing the braid relation (2.4), and then asserts a weight comparison w_{|c_{q-1}|}(δ)=w_{|c_{q-1}|}(γ)−1. No details of this braid-relation manipulation are given, and because the sign condition in the lemma is currently inconsistent, the claimed strict decrease is not demonstrated. This is the branch that must establish well-foundedness for the induction, so a complete, correct verification is required.
minor comments (5)
- [Abstract and Section 1] There are typographical and grammatical errors: 'meth od' should be 'method', and 'As by-product a new complete link invariant are obtained' should be 'As a by-product, a new complete link invariant is obtained.'
- [Algorithm 3.4, Step 1] The tail of the rotated representative is written as 'b1, . . . , bk−1', which appears to be a typo for 'b1, . . . , b_{m−1}'. If m=3 and k=5, the rotation should end with b1,b2, not b1,...,b4.
- [Definition 2.2(5)] The condition 'bq = −cq < 0' is confusing; it should be stated as 'bq = −cq and bq < 0 < cq', since the intended meaning is that when the first sign difference occurs, the negative entry is smaller.
- [Section 4] The sentence 'The complicities is about 1.42c' should read 'The complexity is about 1.42^c'. Also, the geometric average over knots with crossing number at most 12 is an empirical observation, not a worst-case complexity bound, and the text should not imply otherwise.
- [Example in Section 3] The worked example jumps between representatives without indicating which step of Algorithm 3.4 or which braid relation is used at each equality. Annotating the steps would make the example much easier to verify.
Circularity Check
No significant circularity: the skein recursion is genuine and self-contained.
full rationale
The paper's central claim is that Algorithm 3.4, combined with the standard skein relation, computes HOMFLY-type polynomials. This is a genuine recursive expansion: at a terminal representative, Theorem 3.6 explicitly constructs two smaller braid representatives by flipping and deleting the distinguished crossing, and the reduction to their polynomials is exactly the skein relation (1.1) applied to the corresponding link diagrams. No fitted parameter is introduced, and no computed polynomial value is fed back as an input. The base values used in the example, P([1;-]) = 1 and P([2;-]) = -ℓm^{-1} - ℓ^{-1}m^{-1}, are standard consequences of the skein relation or externally accepted facts, not bespoke assumptions tuned to the target knots. The 'new complete link invariant' of Theorem 2.3 is definitional in character, since the minimal representative is a canonical selector of the braid-equivalence classes that are declared to be links; however, this byproduct is not load-bearing for the polynomial computation. The principal genuine weakness is that Lemma 3.5's termination is asserted rather than proved case by case, and the global claim that only finitely many representatives are smaller is false; these are correctness and rigor concerns, not circularity. There are no self-citations carrying the argument, no ansatz smuggled in through the author's prior work, and no prediction that reduces by construction to a fitted quantity.
Assumptions & free parameters
assumptions (5)
- domain assumption The equivalence relations (1) to (5) of Definition 3.1 generate exactly the same-closed-link equivalence on braid representatives (Markov's theorem and conjugacy)
- domain assumption The skein relation (1.1) defines a well-defined link invariant (existence of the HOMFLY polynomial)
- ad hoc to paper The order <br> is well-founded on braid representatives, so minima and descending induction are valid
- domain assumption For a closed braid, flipping the sign of one generator realizes the crossing change L+ to L-, and deleting the generator realizes the oriented smoothing L0
- standard math Artin braid relations (2.3) and (2.4): commutation for far-apart generators and σiσi+1σi = σi+1σiσi+1
Cite this review
Pith. "Pith review of A simple algorithm to compute link polynomials defined by using skein relations." pith.science (2026). https://pith.science/paper/SQNNOBD5
@misc{pith2026190804292,
author = {Pith},
title = {Pith review of: A simple algorithm to compute link polynomials defined by using skein relations},
year = {2026},
howpublished = {\url{https://pith.science/paper/SQNNOBD5}},
note = {Machine review of arXiv:1908.04292}
}
read the original abstract
We give a simple and practical algorithm to compute the link polynomials, which are defined according to the skein relations. Our method is based on a new total order on the set of all braid representatives. As by-product a new complete link invariant are obtained.
Reference graph
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Reviewed August 14, 2026 · model on record in the stance chip above.
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