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Knot Cabling and the Degree of the Colored Jones Polynomial

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Efstratia Kalfagianni, Anh T. Tran

Geometric Topology · Submitted 7 Jan 2015

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

The authors' main claim We show that, under certain hypothesis on this degree, if a knot $K$ satisfies the Slope Conjecture then a $(p, q)$-cable of $K$ satisfies the conjecture, provided that $p/q$ is not a Jones slope of $K$.

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arXiv ID1501.01574 (version 3)
Pith IDUUZWQWCE, Pith's permanent record of this paper and its review
CitationAnh T. Tran, Efstratia Kalfagianni. "Knot Cabling and the Degree of the Colored Jones Polynomial." arXiv:1501.01574 (2015). https://pith.science/paper/UUZWQWCE
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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,472 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.80 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.15 · compares with prior results 0.12 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v3. Results and the main claim are underlined; .

We study the behavior of the degree of the colored Jones polynomial and the boundary slopes of knots under the operation of cabling. We show that, under certain hypothesis on this degree, if a knot $K$ satisfies the Slope Conjecture then a $(p, q)$-cable of $K$ satisfies the conjecture, provided that $p/q$ is not a Jones slope of $K$. As an application we prove the Slope Conjecture for iterated cables of adequate knots and for iterated torus knots. Furthermore we show that, for these knots, the degree of the colored Jones polynomial also determines the topology of a surface that satisfies the Slope Conjecture. We also state a conjecture suggesting a topological interpretation of the linear terms of the degree of the colored Jones polynomial (Conjecture \ref{conj}), and we prove it for the following classes of knots:iterated torus knots and iterated cables of adequate knots, iterated cables of several non-alternating knots with up to nine crossings, pretzel knots of type $(-2, 3, p)$ and their cables, and two-fusion knots.

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 study the behavior of the degree of the colored Jones polynomial and the boundary slopes of knots under the operation of cabling. contribution 0.87 0.85 0.06 asserted 0.65
s1 We show that, under certain hypothesis on this degree, if a knot $K$ satisfies the Slope Conjecture then a $(p, q)$-cable of $K$ satisfies the conjecture, provided that $p/q$ is not a Jones slope of $K$. result 0.96 0.94 0.48 asserted 1.00
s2 As an application we prove the Slope Conjecture for iterated cables of adequate knots and for iterated torus knots. result 0.94 0.96 0.31 asserted 1.00
s3 Furthermore we show that, for these knots, the degree of the colored Jones polynomial also determines the topology of a surface that satisfies the Slope Conjecture. result 0.91 0.95 0.14 asserted 1.00
s4 We also state a conjecture suggesting a topological interpretation of the linear terms of the degree of the colored Jones polynomial (Conjecture \ref{conj}), and we prove it for the following classes of knots:iterated torus knots and iterated cables of adequate knots, iterated cables of several non-alternating knots with up to nine crossings, pretzel knots of type $(-2, 3, p)$ and their cables, and two-fusion knots. result 0.48 0.95 0.16 asserted 0.69

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

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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 study the behavior of the degree of the colored Jones polynomial and the boundary slopes of knots under the operation of cabling. We show that, under certain hypothesis on this degree, if a knot $K$ satisfies the Slope Conjecture then a $(p, q)$-cable of $K$ satisfies the conjecture, provided that $p/q$ is not a Jones slope of $K$. As an application we prove the Slope Conjecture for iterated cables of adequate knots and for iterated torus knots. Furthermore we show that, for these knots, the degree of the colored Jones polynomial also determines the topology of a surface that satisfies the Slope Conjecture. We also state a conjecture suggesting a topological interpretation of the linear terms of the degree of the colored Jones polynomial (Conjecture \ref{conj}), and we prove it for the following classes of knots:iterated torus knots and iterated cables of adequate knots, iterated cables of several non-alternating knots with up to nine crossings, pretzel knots of type $(-2, 3, p)$ and their cables, and two-fusion knots.

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2 arXiv papers cite this in 8 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

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What it cites · 9 sentences without a confident label

Cites as background 2 papers
  • The family of 2-fusion knots is a two-parameter family of closed 3-braids denoted by [equation] For the precise definition and description see [ref] .

    Quadratic integer programming and the slope conjecture · Two-fusion knots · 0.97

  • The reader is referred to Section 3 below for the precise definition (Definition [ref]) and to [ref] and references therein for more details and examples of semi-adequate knots.

    Hyperbolic semi-adequate links · Cable knots · 0.98

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