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One-cocycle invariants for closed braids

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Thomas Fiedler

Geometric Topology · Submitted 9 Apr 2018

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

The authors' main claim We give examples which show that a natural refinement of our invariants can detect (even with linear complexity with respect to the braid length if the number of strands is fixed, and with quadratic complexity if it is not fixed) the non-invertibility of the 2-component link $\hat \beta \cup L\hookrightarrow S^3$, what quantum invariants fail to do.

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arXiv ID1804.03549 (version 1)
Pith IDWT2CG2JM, Pith's permanent record of this paper and its review
CitationThomas Fiedler. "One-cocycle invariants for closed braids." arXiv:1804.03549 (2018). https://pith.science/paper/WT2CG2JM
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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, 4,413 input tokens.

What the abstract itself states, as probabilities: claims first 0.08 · states limitation 0.17 · human participants 0.01 · controlled comparison 0.12 · randomized assignment 0.01 · states quantitative result 0.62 · compares with prior results 0.94 · mentions code or data availability 0.02.

The abstract, sentence by sentence

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

We introduce new polynomial isotopy invariants for closed braids. They are constructed as polynomial valued {\em Gauss diagram 1-cocycles} evaluated on the full rotation of the closed braid $\hat \beta$ around the core of the corresponding solid torus. They can be calculated with polynomial complexity with respect to the braid length and their derivatives evaluated at $x=1$ are finite type invariants of closed braids. Let the solid torus V be standardly embedded in the 3-sphere and let L be the core of the complementary solid torus $S^3\setminus V$. We give examples which show that a natural refinement of our invariants can detect (even with linear complexity with respect to the braid length if the number of strands is fixed, and with quadratic complexity if it is not fixed) the non-invertibility of the 2-component link $\hat \beta \cup L\hookrightarrow S^3$, what quantum invariants fail to do.

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 introduce new polynomial isotopy invariants for closed braids. contribution 1.00 0.95 0.47 asserted 1.00
s1 They are constructed as polynomial valued {\em Gauss diagram 1-cocycles} evaluated on the full rotation of the closed braid $\hat \beta$ around the core of the corresponding solid torus. method 0.91 0.90 0.21 asserted 1.00
s2 They can be calculated with polynomial complexity with respect to the braid length and their derivatives evaluated at $x=1$ are finite type invariants of closed braids. result 0.78 0.85 0.25 asserted 0.92
s3 Let the solid torus V be standardly embedded in the 3-sphere and let L be the core of the complementary solid torus $S^3\setminus V$. background 0.88 0.15 0.02 asserted 0.82
s4 We give examples which show that a natural refinement of our invariants can detect (even with linear complexity with respect to the braid length if the number of strands is fixed, and with quadratic complexity if it is not fixed) the non-invertibility of the 2-component link $\hat \beta \cup L\hookrightarrow S^3$, what quantum invariants fail to do. result 0.93 0.93 0.39 asserted 1.00

What kind of paper the classifier reports this is: method or tool 0.64 · evidence from proof or derivation 0.97.

What was examined, and what was not

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  • full text
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  • references and citing sentences
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  • P abstract
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  • 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 introduce new polynomial isotopy invariants for closed braids. They are constructed as polynomial valued {\em Gauss diagram 1-cocycles} evaluated on the full rotation of the closed braid $\hat \beta$ around the core of the corresponding solid torus. They can be calculated with polynomial complexity with respect to the braid length and their derivatives evaluated at $x=1$ are finite type invariants of closed braids. Let the solid torus V be standardly embedded in the 3-sphere and let L be the core of the complementary solid torus $S^3\setminus V$. We give examples which show that a natural refinement of our invariants can detect (even with linear complexity with respect to the braid length if the number of strands is fixed, and with quadratic complexity if it is not fixed) the non-invertibility of the 2-component link $\hat \beta \cup L\hookrightarrow S^3$, what quantum invariants fail to do.

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