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Rectangle condition and its applications

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Bo-hyun Kwon

Geometric Topology · Submitted 28 Apr 2014

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

The authors' main claim We show that the satisfaction of the rectangle condition for a $n$-bridge decomposition can guarantee that the Hempel distance for the $n$-bridge decomposition is greater than or equal to $2$.

how it is stated · asserted 0.98 read the record →

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arXiv ID1404.7165 (version 6)
Pith IDTUBCZX5F, Pith's permanent record of this paper and its review
CitationBo-hyun Kwon. "Rectangle condition and its applications." arXiv:1404.7165 (2014). https://pith.science/paper/TUBCZX5F
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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, 3,011 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.06 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.96 · compares with prior results 0.16 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v6. Results and the main claim are underlined; .

In this paper, we define the rectangle condition on the bridge sphere for a $n$-bridge decomposition of a knot whose definition is analogous to the definition of the rectangle condition for Heegaard splittings of $3$-manifolds. We show that the satisfaction of the rectangle condition for a $n$-bridge decomposition can guarantee that the Hempel distance for the $n$-bridge decomposition is greater than or equal to $2$. In particular, we give an interesting family of alternating 3-bridge knots by using the rectangle condition and a modified train track argument.

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 In this paper, we define the rectangle condition on the bridge sphere for a $n$-bridge decomposition of a knot whose definition is analogous to the definition of the rectangle condition for Heegaard splittings of $3$-manifolds. contribution 0.97 0.91 0.05 asserted 0.99
s1 We show that the satisfaction of the rectangle condition for a $n$-bridge decomposition can guarantee that the Hempel distance for the $n$-bridge decomposition is greater than or equal to $2$. result 0.99 0.95 0.69 asserted 0.98
s2 In particular, we give an interesting family of alternating 3-bridge knots by using the rectangle condition and a modified train track argument. contribution 0.82 0.95 0.11 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.99 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read
  • arXiv listing abstract v6
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

In this paper, we define the rectangle condition on the bridge sphere for a $n$-bridge decomposition of a knot whose definition is analogous to the definition of the rectangle condition for Heegaard splittings of $3$-manifolds. We show that the satisfaction of the rectangle condition for a $n$-bridge decomposition can guarantee that the Hempel distance for the $n$-bridge decomposition is greater than or equal to $2$. In particular, we give an interesting family of alternating 3-bridge knots by using the rectangle condition and a modified train track argument.

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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

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