Khovanov homology of tangles: algorithm and computation

Guo-Wei Wei; Jian Liu; Li Shen

REVIEW 4 major objections 3 minor 17 references

Khovanov homology of tangles: algorithm and computation

T0 review · 4 major / 3 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read This paper gives a direct, code-backed procedure for computing Khovanov homology of tangles from a diagram.

desk verdict Unreadable full text and no code make the algorithm impossible to assess; a clean version with validation could be a useful computational companion, but the current submission is not refereeable. read the letter →

arxiv 2508.14404 v1 pith:F2UZC3A6 submitted 2025-08-20 math.GT

Li Shen , Jian Liu , Guo-Wei Wei This is my paper · ORCID

classification math.GT MSC 57K1857K1057R56

keywords KhovanovhomologytanglesTQFTknotinvariantscomputationaltopologychaincomplexalgorithm

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

Khovanov homology is a powerful invariant of knots and links, but until now its computation has been largely limited to closed knots and links. Tangles, which are knot-like diagrams whose strands may have open ends, are more flexible and appear naturally in biology, chemistry, and quantum physics. The paper claims that a recent topological quantum field theory (TQFT) construction for tangle Khovanov homology can be turned into a concrete, step-by-step algorithm. It also provides implementation code, so that a researcher can in principle compute the Khovanov homology of a tangle directly from its diagram. If the procedure is correct, it extends a standard tool of knot theory to a much broader class of objects.

What carries the argument

The load-bearing object is the TQFT construction that assigns a chain complex to a tangle with free endpoints on its boundary. It makes the computation local: the algorithm decomposes the tangle diagram into pieces, applies the TQFT's maps piece by piece, and assembles the results into a single chain complex. The code is the practical implementation of that assembly.

What would settle it

Run the algorithm on two diagrams of the same tangle that differ by a Reidemeister move; if the resulting homology groups differ, the claimed invariant fails. Alternatively, close a tangle into a knot and compare the computed groups with the standard Khovanov homology of that closure computed by established software; a mismatch in any rank or grading would refute the construction.

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Extended reading notes

Core claim

The paper's central claim is that the authors' earlier TQFT construction for Khovanov homology of tangles can be made algorithmic. Given a tangle diagram, the procedure produces homology groups by following the TQFT's local rules, assembling a chain complex whose homology is the tangle invariant. The paper describes the computation in detail and supplies code to carry it out. For closed knots and links, the construction is designed to reproduce the standard Khovanov homology, so the tangle computation is an extension of the familiar invariant rather than a replacement.

Load-bearing premise

The algorithm's outputs count as tangle Khovanov homology only if the earlier TQFT construction is a true invariant—independent of how the tangle is drawn—and agrees with ordinary Khovanov homology once the open ends are closed up. The present paper supplies the computation procedure, not the proof of that foundation.

Editorial extensions

If this is right

Anyone with a tangle diagram can in principle compute its Khovanov homology mechanically, without designing a bespoke spectral sequence.
The procedure turns tangle Khovanov homology into a computational tool for studying protein, DNA, and other molecular structures naturally modeled by tangles.
Automated computation makes it feasible to generate tables of tangle homology, which can be mined for patterns and conjectures.
Because the construction is a TQFT, the same code can be extended to compute maps induced by tangle cobordisms, not just the homology groups.

Reading between the lines

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

A natural next step is to benchmark the code against the closure test: closing a tangle into a knot or link should recover ordinary Khovanov homology up to a grading shift, providing an independent check of the implementation.
The same algorithmic skeleton should transfer to other Khovanov-type theories, such as sl(3) or annular Khovanov homology, wherever a TQFT description exists; the paper itself only claims the Khovanov case.
The practical ceiling will likely mirror that of knot Khovanov homology: exponential growth in the number of crossings. The paper's code provides a platform for measuring that growth on concrete tangles.

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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 honest bottom line: I can't review this paper, and neither can a referee. The supplied full text is corrupted mojibake with a different arXiv ID embedded, so the algorithm, its pseudocode, worked examples, and any validation tables are unreadable. Judging only from the abstract, the claimed contribution is a practical computation procedure and code guide for tangle Khovanov homology built on the authors' own TQFT construction. That is a reasonable thing to want: tangle invariants matter for DNA/protein topology and physics, and a usable implementation would be a service to that community. The abstract is also honest in framing this as a companion to the prior TQFT paper, not as a new invariant.

The soft spots are load-bearing but mostly not ones I can check. First, the correctness of every computed table depends on the prior TQFT construction, which is only referenced as 'our recent work' with no citation detail, no proof summary, and no verification here. That is circular if the reader is expected to take the tables as evidence for the construction. A companion paper can get away with this if the base paper is published and the code is checkable; neither condition is met. Second, the abstract says tangle Khovanov computation is 'largely unexplored,' which oversells the situation—there is existing literature on functor-valued tangle invariants. Third, the paper promises code but provides no artifact; without code or detailed reproducible examples, 'practical guide' is not yet a deliverable. I am not claiming the mathematics is wrong. I cannot tell. But the central claim—that a researcher can implement and verify the procedure from this paper—is unsupported by the supplied text.

Recommendation: desk-reject the current version, not because the idea is bad, but because the submission is unreadable and missing its key artifact. Ask the authors to resubmit with a clean PDF, a precise citation to the TQFT construction, the code repository, and validation of their outputs against known Khovanov homology for closures or small knotted tangles. If they supply those, I'd take another look.

Referee Report

4 major / 3 minor

Summary. This manuscript (arXiv:2508.14404) purports to give a practical algorithm for computing Khovanov homology of tangles, building on a TQFT construction the authors say they introduced in recent work. The abstract states that the paper provides a 'detailed computation procedure' and 'a practical guide for implementing algorithms through codes.' The supplied full text, however, is not readable: it consists largely of mojibake, contains an embedded arXiv identifier for an unrelated cs.CV paper (arXiv:2508.14405v1), and does not expose any pseudocode, worked example, theorem statement, or validation table in intelligible form. I therefore could not verify the existence or correctness of the claimed algorithm, its implementation, or its agreement with known Khovanov homology computations. The paper's central claim—that this is a comprehensive, implementable computational method—is currently unsupported by the submitted text.

Significance. If the algorithm and the underlying TQFT construction are correct, this would be a useful contribution: tangle Khovanov homology is much less developed computationally than the knot/link case, and a concrete procedure with code would open applications in biology and chemistry. The stated goal is valuable and timely. However, as submitted, the paper delivers no checkable content: no explicit algorithm description, no reproducible code, no comparison with the standard Khovanov homology for closed knots/links, and no reference or proof establishing that the prior TQFT is invariant under Reidemeister moves. The contribution therefore cannot currently be used by readers, and its significance cannot be assessed. I would welcome a clean resubmission in which the algorithm, the TQFT foundations, the code, and validation are verifiable.

major comments (4)

[Full text (entire manuscript)] The supplied text is corrupted and unreadable; it includes the unrelated line 'arXiv:2508.14405v1 [cs.CV] 20 Aug 2025' and otherwise consists largely of mojibake. No pseudocode, worked example, or theorem/proof can be extracted. This blocks verification of the claimed 'detailed computation procedure'; condition (1) of the abstract's central claim is unsupported.
[Abstract, first paragraph] The computed object is defined by 'our recent work'—a TQFT construction that is neither cited nor reproduced here. The present paper therefore does not establish that the algorithm computes a well-defined invariant: any failure of Reidemeister invariance or functoriality in the prior construction propagates into every output. The paper is not self-contained against an external standard (e.g., agreement with standard Khovanov homology for closed links), so the computations cannot be validated from the submitted text.
[Abstract and conclusion] The abstract promises 'a practical guide for implementing algorithms through codes,' but the supplied text contains no code, no runnable examples, and no output tables. Thus the claimed practical computational contribution is not testable. I could not confirm even a single sample computation against known values.
[Validation] No validation against standard Khovanov homology is visible in the corrupted text. Since tangle invariants specialize to knot/link invariants by closing the tangle, a few closed tangle examples compared with known knot/link Khovanov homology would be a decisive check. As submitted, this check is absent (or unreadable), leaving the correctness of the algorithm unestablished.

minor comments (3)

[File encoding/header] The embedded unrelated arXiv identifier must be removed; it is evidence of file corruption. A clean resubmission should ensure correct font encoding so that all mathematical symbols and diagrams are readable.
[References] The 'recent work' containing the TQFT construction should be cited explicitly in the abstract and introduction, or summarized in an appendix, so that the dependency is transparent.
[Notation and examples] Because of the corrupted text, notation and examples could not be checked. A resubmission should include a clear statement of the smoothing calculus, edge assignment, and differential, each illustrated on a small tangle.

Circularity Check

1 steps flagged · score 4.0 of 10

Central computation rests on the authors' own prior TQFT construction, which is never independently verified in the manuscript.

self citation load bearing [Abstract]
"In our recent work, we provide a topological quantum field theory (TQFT) construction for the Khovanov homology of tangles, offering a more concrete method for its computation. The primary contribution of this work is a comprehensive approach to the computation of the Khovanov homology of tangles..."

The object whose computation is promised is not independently defined or benchmarked in this paper: it is the output of a TQFT construction that the authors attribute to their own 'recent work.' No proof is given here that this TQFT agrees with any external notion of Khovanov homology for tangles, and no Reidemeister-invariance check or comparison to known examples is presented in the readable text. Consequently, every computed table is valid only if the unexhibited self-cited construction is correct; the paper's central claim reduces to accepting that self-citation as the definition/construction of the invariant. This is load-bearing self-citation rather than an independent derivation.

full rationale

The abstract contains one clearly circular dependency: the algorithm's target is the Khovanov homology of tangles as produced by the authors' own prior TQFT construction, and no independent verification of that construction is supplied. The full text is severely corrupted (e.g., an unrelated arXiv identifier 'arXiv:2508.14405v1 [cs.CV]' appears mid-manuscript), so I could not locate any additional derivations, equations, or benchmarks to check. Despite this, no further fitted-input predictions, ansatz-smuggling, or renaming patterns are visible in the readable portions. The self-citation is load-bearing for correctness, but the paper may still contain independent algorithmic content (a computation procedure and code guide), so the circularity is partial rather than total. Score 4 reflects 'some self-citation; central claim still has independent content.'

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The central claim is an algorithm for a quantity defined by the authors' own prior TQFT paper; every computed output inherits the correctness of that construction. No independent benchmarks, datasets, or external baselines are visible at the abstract level; validation would have to come from matching standard Khovanov homology on closed knots and links, which is not mentioned.

assumptions (3)

domain assumption The TQFT construction for Khovanov homology of tangles from the authors' prior work is correct and is a genuine invariant.
The present paper's computation is built on this construction (abstract: 'In our recent work, we provide a TQFT construction...'). If this construction is wrong, the computed outputs are not Khovanov homology.
domain assumption The TQFT functor restricts to standard Khovanov homology on closed knots and links.
Implied by calling the object 'the Khovanov homology of tangles'; needed so that computations can be checked against known knot/link results. Not stated in the abstract.
standard math Standard algebraic tools (chain complexes, homology over Z or Q, Temperley-Lieb basis for tangle skein modules) behave as usual.
Any concrete algorithm for Khovanov homology assumes the standard categorical and algebraic toolkit of the subject.

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Cite this review

Pith. "Pith review of Khovanov homology of tangles: algorithm and computation." pith.science (2026). https://pith.science/paper/F2UZC3A6

@misc{pith2026250814404,
  author       = {Pith},
  title        = {Pith review of: Khovanov homology of tangles: algorithm and computation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/F2UZC3A6}},
  note         = {Machine review of arXiv:2508.14404}
}

read the original abstract

Knot, link, and tangle theory is crucial in both mathematical theory and practical application, including quantum physics, molecular biology, and structural chemistry. Unlike knots and links, tangles impose more relaxed constraints, allowing the presence of arcs, which makes them particularly valuable for broader applications. Although Khovanov homology for knots and links has been extensively studied, its computation for tangles remains largely unexplored. In our recent work, we provide a topological quantum field theory (TQFT) construction for the Khovanov homology of tangles, offering a more concrete method for its computation. The primary contribution of this work is a comprehensive approach to the computation of the Khovanov homology of tangles, offering both a detailed computation procedure and a practical guide for implementing algorithms through codes to facilitate the calculation. This contribution paves the way for further studies and applications of Khovanov homology in the context of tangles.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

17 extracted references · 14 canonical work pages

[1]

The knot book: an elementary introduction to the mathematical theory of knots

Colin Conrad Adams. The knot book: an elementary introduction to the mathematical theory of knots . American Mathematical Soc., 2004

2004
[2]

On K hovanov's categorification of the J ones polynomial

Dror Bar-Natan. On K hovanov's categorification of the J ones polynomial. Algebraic & Geometric Topology , 2(1):337--370, 2002

work page 2002
[3]

Khovanov's homology for tangles and cobordisms

Dror Bar-Natan. Khovanov's homology for tangles and cobordisms. Geometry & Topology , 9(3):1443--1499, 2005

work page 2005
[4]

Fast K hovanov homology computations

Dror Bar-Natan. Fast K hovanov homology computations. Journal of Knot Theory and Its Ramifications , 16(03):243--255, 2007

work page 2007
[5]

Conformational parameters for amino acids in helical, -sheet, and random coil regions calculated from proteins

Peter Y Chou and Gerald D Fasman. Conformational parameters for amino acids in helical, -sheet, and random coil regions calculated from proteins. Biochemistry , 13(2):211--222, 1974

work page 1974
[6]

The protein-folding problem, 50 years on

Ken A Dill and Justin L MacCallum. The protein-folding problem, 50 years on. science , 338(6110):1042--1046, 2012

work page 2012
[7]

Molecular knots

Stephen DP Fielden, David A Leigh, and Steffen L Woltering. Molecular knots. Angewandte Chemie International Edition , 56(37):11166--11194, 2017

work page 2017
[8]

Chemical topology: complex molecular knots, links, and entanglements

Ross S Forgan, Jean-Pierre Sauvage, and J Fraser Stoddart. Chemical topology: complex molecular knots, links, and entanglements. Chemical Reviews , 111(9):5434--5464, 2011

work page 2011

Show all 17 references

[9]

Quantum field theory and knot invariants

Chun-Chung Hsieh. Quantum field theory and knot invariants. In Advances in Quantum Field Theory , pages 155--184. IntechOpen, 2012

2012
[10]

A categorification of the J ones polynomial

Mikhail Khovanov. A categorification of the J ones polynomial. Duke Math. J. , 104(1):359--426, 2000

2000
[11]

A functor-valued invariant of tangles

Mikhail Khovanov. A functor-valued invariant of tangles. Algebraic & Geometric Topology , 2(2):665--741, 2002

2002
[12]

Persistent K hovanov homology of tangles

Jian Liu, Li Shen, and Guo-Wei Wei. Persistent K hovanov homology of tangles. arXiv preprint arXiv:2409.18312 , 2024

2024
[13]

Knots and links

Dale Rolfsen. Knots and links . Number 346. American Mathematical Soc., 2003

2003
[14]

A quantum algorithm for K hovanov homology

Alexander Schmidhuber, Michele Reilly, Paolo Zanardi, Seth Lloyd, and Aaron Lauda. A quantum algorithm for K hovanov homology. arXiv preprint arXiv:2501.12378 , 2025

2025 arXiv
[15]

Evolutionary K hovanov homology

Li Shen, Jian Liu, and Guo-Wei Wei. Evolutionary K hovanov homology. AIMS Mathematics , 9(9):26139--26165, 2024

2024
[16]

Computations in K hovanov homology

Brandon Williams. Computations in K hovanov homology. University of Illinois at Chicago, Chicago, Illinois , 2008

2008
[17]

Quantum field theory and the J ones polynomial

Edward Witten. Quantum field theory and the J ones polynomial. Communications in Mathematical Physics , 121(3):351--399, 1989

1989

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [1]

The knot book: an elementary introduction to the mathematical theory of knots

Colin Conrad Adams. The knot book: an elementary introduction to the mathematical theory of knots . American Mathematical Soc., 2004

2004

[2] [2]

On K hovanov's categorification of the J ones polynomial

Dror Bar-Natan. On K hovanov's categorification of the J ones polynomial. Algebraic & Geometric Topology , 2(1):337--370, 2002

work page 2002

[3] [3]

Khovanov's homology for tangles and cobordisms

Dror Bar-Natan. Khovanov's homology for tangles and cobordisms. Geometry & Topology , 9(3):1443--1499, 2005

work page 2005

[4] [4]

Fast K hovanov homology computations

Dror Bar-Natan. Fast K hovanov homology computations. Journal of Knot Theory and Its Ramifications , 16(03):243--255, 2007

work page 2007

[5] [5]

Conformational parameters for amino acids in helical, -sheet, and random coil regions calculated from proteins

Peter Y Chou and Gerald D Fasman. Conformational parameters for amino acids in helical, -sheet, and random coil regions calculated from proteins. Biochemistry , 13(2):211--222, 1974

work page 1974

[6] [6]

The protein-folding problem, 50 years on

Ken A Dill and Justin L MacCallum. The protein-folding problem, 50 years on. science , 338(6110):1042--1046, 2012

work page 2012

[7] [7]

Molecular knots

Stephen DP Fielden, David A Leigh, and Steffen L Woltering. Molecular knots. Angewandte Chemie International Edition , 56(37):11166--11194, 2017

work page 2017

[8] [8]

Chemical topology: complex molecular knots, links, and entanglements

Ross S Forgan, Jean-Pierre Sauvage, and J Fraser Stoddart. Chemical topology: complex molecular knots, links, and entanglements. Chemical Reviews , 111(9):5434--5464, 2011

work page 2011