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A few weight systems arising from intersection graphs

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Blake Mellor

Geometric Topology · Submitted 12 Apr 2000

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

The authors' main claim We show that the adjacency matrices of the intersection graphs of chord diagrams satisfy the 2-term relations of Bar-Natan and Garoufalides [bg], and hence give rise to weight systems.

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arXiv IDmath/0004080 (version 3)
Pith IDLS7HURGO, Pith's permanent record of this paper and its review
CitationBlake Mellor. "A few weight systems arising from intersection graphs." arXiv:math/0004080 (2000). https://pith.science/paper/LS7HURGO
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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-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,593 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.06 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.05 · compares with prior results 0.50 · mentions code or data availability 0.02.

The abstract, sentence by sentence

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

We show that the adjacency matrices of the intersection graphs of chord diagrams satisfy the 2-term relations of Bar-Natan and Garoufalides [bg], and hence give rise to weight systems. Among these weight systems are those associated with the Conway and HOMFLYPT polynomials. We extend these ideas to looking at a space of {\it marked} chord diagrams modulo an extended set of 2-term relations, define a set of generators for this space, and again derive weight systems from the adjacency matrices of the (marked) intersection graphs. Among these weight systems are those associated with the Kauffman polynomial.

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 show that the adjacency matrices of the intersection graphs of chord diagrams satisfy the 2-term relations of Bar-Natan and Garoufalides [bg], and hence give rise to weight systems. result 0.65 0.93 0.75 asserted 1.00
s1 Among these weight systems are those associated with the Conway and HOMFLYPT polynomials. result 0.95 0.49 0.09 asserted 1.00
s2 We extend these ideas to looking at a space of {\it marked} chord diagrams modulo an extended set of 2-term relations, define a set of generators for this space, and again derive weight systems from the adjacency matrices of the (marked) intersection graphs. contribution 0.99 0.96 0.24 asserted 1.00
s3 Among these weight systems are those associated with the Kauffman polynomial. result 0.99 0.83 0.12 asserted 1.00

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

What was examined, and what was not

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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 show that the adjacency matrices of the intersection graphs of chord diagrams satisfy the 2-term relations of Bar-Natan and Garoufalides [bg], and hence give rise to weight systems. Among these weight systems are those associated with the Conway and HOMFLYPT polynomials. We extend these ideas to looking at a space of {\it marked} chord diagrams modulo an extended set of 2-term relations, define a set of generators for this space, and again derive weight systems from the adjacency matrices of the (marked) intersection graphs. Among these weight systems are those associated with the Kauffman polynomial.

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