Almost positive links are strongly quasipositive

Andrew Lobb; Lukas Lewark; Peter Feller

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Almost positive links are strongly quasipositive

Peter Feller, Lukas Lewark, Andrew Lobb

Geometric Topology · Submitted 18 Sep 2018

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

The authors' main claim We prove that any link admitting a diagram with a single negative crossing is strongly quasipositive.

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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-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,611 input tokens.

What the abstract itself states, as probabilities: claims first 0.07 · states limitation 0.28 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.64 · compares with prior results 0.42 · mentions code or data availability 0.02.

The abstract, sentence by sentence

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

We prove that any link admitting a diagram with a single negative crossing is strongly quasipositive. This answers a question of Stoimenow's in the (strong) positive. As a second main result, we give simple and complete characterizations of link diagrams with quasipositive canonical surface (the surface produced by Seifert's algorithm). As applications, we determine which prime knots up to 13 crossings are strongly quasipositive, and we confirm the following conjecture for knots that have a canonical surface realizing their genus: a knot is strongly quasipositive if and only if the Bennequin inequality is an equality.

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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 prove that any link admitting a diagram with a single negative crossing is strongly quasipositive.	result 0.96	0.96	0.89	asserted 1.00
s1	This answers a question of Stoimenow's in the (strong) positive.	interpretation 0.64	0.65	0.09	asserted 1.00
s2	As a second main result, we give simple and complete characterizations of link diagrams with quasipositive canonical surface (the surface produced by Seifert's algorithm).	result 0.58	0.96	0.14	asserted 1.00
s3	As applications, we determine which prime knots up to 13 crossings are strongly quasipositive, and we confirm the following conjecture for knots that have a canonical surface realizing their genus: a knot is strongly quasipositive if and only if the Bennequin inequality is an equality.	result 0.78	0.94	0.19	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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arXiv listing abstract v3

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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 prove that any link admitting a diagram with a single negative crossing is strongly quasipositive. This answers a question of Stoimenow's in the (strong) positive. As a second main result, we give simple and complete characterizations of link diagrams with quasipositive canonical surface (the surface produced by Seifert's algorithm). As applications, we determine which prime knots up to 13 crossings are strongly quasipositive, and we confirm the following conjecture for knots that have a canonical surface realizing their genus: a knot is strongly quasipositive if and only if the Bennequin inequality is an equality.

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References

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2 arXiv papers cite this in 4 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

Papers citing it · 3 sentences without a confident label

Cites it as background 1 paper

Shortly before publicizing this paper, the authors learned that in [ref] , Feller, Lewark, and Lobb prove that almost positive links are strongly quasipositive.

Positivities of knots and links and the defect of Bennequin inequality · Relations among positivities and summary of our results · 0.89

What it cites · 10 sentences without a confident label

Uses the method or tool of 2 papers

The quasipositivity of links admitting diagrams of type II can be shown using a method due to Baader [ref] as remarked by Tagami [ref] .

Slice and Gordian numbers of track knots · Approach to proofs · 1.00
The quasipositivity of links admitting diagrams of type II can be shown using a method due to Baader [ref] as remarked by Tagami [ref] .

The Rasmussen invariant, four-genus and three-genus of an almost positive knot are equal · Approach to proofs · 1.00

Cites as background 6 papers

The distinction between type I and II is rather natural and has been made previously [ref] .

The Rasmussen invariant, four-genus and three-genus of an almost positive knot are equal · Approach to proofs · 0.83
The distinction between type I and II is rather natural and has been made previously [ref] .

On polynomials and surfaces of variously positive links · Approach to proofs · 0.83
For example, Cromwell showed that almost positive links have Conway polynomials with non-negative coefficients [ref] ; Przytycki and Taniyama proved they have negative signature [ref] ; Stoimenow showed that non-trivial almost positive links are chiral and non-slice [ref] ; and Tagami proved that the 3–genus, 4–genus, and (normalized) Rasmussen invariant of almost positive knots agree [ref] . <ref> can be seen in this context.

Almost positive links have negative signature · Main results · 0.84

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