Properties of knots preserved by cabling

Alexander Zupan

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Properties of knots preserved by cabling

Alexander Zupan

Geometric Topology · Submitted 15 Oct 2010

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

The authors' main claim We use this information to demonstrate that thin position is a minimal bridge position of J if and only if the same is true for K, and more generally we show that any thin position of K is an "obvious" cabling of a thin position of J. We conclude by proving that J is meridionally small (mp-small) if and only if K is meridionally small (mp-small), and if J is mp-small and every non-minimal bridge position of J is stabilized, then the same is true for K.

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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,484 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.13 · human participants 0.01 · controlled comparison 0.06 · randomized assignment 0.01 · states quantitative result 0.10 · compares with prior results 0.10 · mentions code or data availability 0.01.

The abstract, sentence by sentence

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

We examine geometric properties of a knot J that are unchanged by taking a (p,q)-cable K of J. Specifically, we relate w(K) to w(J), where w(K) is the width of K in the sense of Gabai. We use this information to demonstrate that thin position is a minimal bridge position of J if and only if the same is true for K, and more generally we show that any thin position of K is an "obvious" cabling of a thin position of J. We conclude by proving that J is meridionally small (mp-small) if and only if K is meridionally small (mp-small), and if J is mp-small and every non-minimal bridge position of J is stabilized, then the same is true for K.

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What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We examine geometric properties of a knot J that are unchanged by taking a (p,q)-cable K of J. Specifically, we relate w(K) to w(J), where w(K) is the width of K in the sense of Gabai.	contribution 0.98	0.88	0.09	asserted 0.92
s1	We use this information to demonstrate that thin position is a minimal bridge position of J if and only if the same is true for K, and more generally we show that any thin position of K is an "obvious" cabling of a thin position of J. We conclude by proving that J is meridionally small (mp-small) if and only if K is meridionally small (mp-small), and if J is mp-small and every non-minimal bridge position of J is stabilized, then the same is true for K.	result 0.95	0.95	0.74	asserted 1.00

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

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The paper

Original abstract

We examine geometric properties of a knot J that are unchanged by taking a (p,q)-cable K of J. Specifically, we relate w(K) to w(J), where w(K) is the width of K in the sense of Gabai. We use this information to demonstrate that thin position is a minimal bridge position of J if and only if the same is true for K, and more generally we show that any thin position of K is an "obvious" cabling of a thin position of J. We conclude by proving that J is meridionally small (mp-small) if and only if K is meridionally small (mp-small), and if J is mp-small and every non-minimal bridge position of J is stabilized, then the same is true for K.

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2 arXiv papers cite this in 2 sentences

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

What it cites · 8 sentences without a confident label

Builds on 2 papers

In order to further develop the connection between bridge number and knot width, we make the following conjecture in [ref] as an analogue to the above theorem of Schubert and Schultens: If $K$ is a satellite knot with companion $J$ and pattern $K̂$ with wrapping number $n$, then [equation] One purpose of this paper is to prove the conjecture for cable knots and to show that the bound is tight, as with bridge number.

A lower bound on the width of satellite knots · Introduction · 0.90
Although [ref] deals specifically with torus knots, some results apply in the setting of mp-small cable knots.

Non-minimal bridge positions of torus knots are stabilized · Destabilizing non-minimal bridge positions of cables · 0.93

Uses the method or tool of 1 paper

By the proof of Lemma 3.3 in [ref] , we can cancel the saddle $σ$ with the maximum contained in $D_1$ without increasing $w(k)$.

A lower bound on the width of satellite knots · Efficient position of $k$ · 0.92

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