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Does the Jones polynomial detect the unknot?

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Paper breakdownThe authors' claims, conditions and measurements, as JSON Signed receiptProof this review has not been altered
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Stephen J. Bigelow

Geometric Topology · Submitted 12 Dec 2000

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim To find such a knot, it is almost certainly sufficient to find a non-trivial braid on four strands in the kernel of the Burau representation.

how it is stated · hedged possibility 0.93 read the record →

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arXiv IDmath/0012086 (version 1)
Pith IDVDTGE2WW, Pith's permanent record of this paper and its review
CitationStephen J. Bigelow. "Does the Jones polynomial detect the unknot?." arXiv:math/0012086 (2000). https://pith.science/paper/VDTGE2WW
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The compiled record

The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,844 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.29 · human participants 0.02 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.05 · compares with prior results 0.05 · mentions code or data availability 0.04.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

We address the question: Does there exist a non-trivial knot with a trivial Jones polynomial? To find such a knot, it is almost certainly sufficient to find a non-trivial braid on four strands in the kernel of the Burau representation. I will describe a computer algorithm to search for such a braid.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We address the question: Does there exist a non-trivial knot with a trivial Jones polynomial? background 0.83 0.08 0.04 unclear 0.91
s1 To find such a knot, it is almost certainly sufficient to find a non-trivial braid on four strands in the kernel of the Burau representation. method 0.66 0.13 0.06 hedged possibility 0.93
s2 I will describe a computer algorithm to search for such a braid. method 0.72 0.64 0.04 asserted 0.97

What kind of paper the classifier reports this is: method or tool 0.99 · evidence from simulation or computation 0.60.

What was examined, and what was not

sources read
  • arXiv listing abstract v1
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

We address the question: Does there exist a non-trivial knot with a trivial Jones polynomial? To find such a knot, it is almost certainly sufficient to find a non-trivial braid on four strands in the kernel of the Burau representation. I will describe a computer algorithm to search for such a braid.

Human review

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References
How the literature uses it

1 arXiv paper cite this in 1 sentence

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it

Cites it as background 1 paper
  • Bigelow gave an influential discussion of the problem and of several equivalent or closely related formulations [ref] ; see also the recent survey of Manolescu [ref] .

    On trivial Jones--Vassiliev polynomials · Introduction · 1.00

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