{"id":"W2950905993","doi":"10.48550/arxiv.math/0110016","title":"On the crossing number of positive knots and braids and braid index criteria of Jones and Morton-Williams-Franks","year":2001,"lang":"en","type":"preprint","venue":"ArXiv.org","topic":"Geometric and Algebraic Topology","field":"Mathematics","cited_by":0,"is_retracted":false,"has_abstract":true,"ca_institutions":"University of Toronto","funders":"","keywords":"Braid; Index (typography); Mathematics; Crossing number (knot theory); History; Knot (papermaking); Computer science; Materials science; Archaeology; Composite material","routes":{"ca_aff":true,"ca_fund":false,"ca_venue":false,"about_ca":false,"invisible_to_affiliation_only":false},"retraction":null,"screen":null,"direct_labels":[],"prediction":{"model_version":"metacan-v3-hybrid-931329e0061c","candidate_categories":[],"consensus_categories":[],"category_scores_codex":[0.001116153,0.0004496595,0.0005185068,0.004736277,0.001136312,0.002184129,0.0006163615,0.0007421842,0.00296567],"category_scores_gemma":[0.007013712,0.0003367041,0.0003843047,0.001795362,0.003769448,0.003162745,0.00185,0.001183678,0.0002942561],"about_ca_system_candidate":false,"about_ca_system_consensus":false,"about_ca_system_score_codex":0.0009564743,"about_ca_system_score_gemma":0.0002177428,"about_ca_topic_candidate":false,"about_ca_topic_consensus":false,"about_ca_topic_score_codex":0.000825076,"about_ca_topic_score_gemma":0.0007478886,"domain_scores_codex":[0.9994209,0.0000899685,0.00003063804,0.0001330853,0.0002205411,0.0001048809],"domain_scores_gemma":[0.9962302,0.002009068,0.0005875073,0.0002630836,0.0003589951,0.0005511811],"domain_codex":null,"domain_gemma":null,"domain_candidate":null,"domain_consensus":null,"study_design_codex":"theoretical_or_conceptual","study_design_gemma":"theoretical_or_conceptual","study_design_scores_codex":[0.00006461178,0.00002148488,0.004645919,0.00007083268,0.000009065607,0.0001698279,0.0005002344,0.003495908,0.002552213,0.9758593,0.001017819,0.01159262],"study_design_scores_gemma":[0.00001252156,0.00005024838,0.004983315,0.00005108794,0.00002268696,0.0006457917,0.0002913095,0.01583774,0.003203944,0.9700874,0.004769528,0.00004443349],"study_design_candidate":"theoretical_or_conceptual","study_design_consensus":"theoretical_or_conceptual","genre_codex":"empirical","genre_gemma":"empirical","genre_scores_codex":[0.8511496,0.001919424,0.06075902,0.0012377,0.0001796427,0.00004075234,0.0001910019,0.000221118,0.08430181],"genre_scores_gemma":[0.9905477,0.0003849732,0.005779132,0.00006088298,0.00008750041,0.00001272738,0.00006173744,0.00002393351,0.003041415],"genre_candidate":"empirical","genre_consensus":"empirical","teacher_disagreement_score":0.004736277,"threshold_uncertainty_score":0.009921134,"prediction_status":"machine_predicted_unvalidated"},"machine_scores":{"provisional":true,"baseline":true,"maturity_gate_passed":false,"score_opus":0.0552909694437595,"score_gpt":0.3439991362638957,"score_spread":0.2887081668201362,"validation_status":"score_only:v0-immature-baseline","note":"Baseline scores from an immature model (maturity gate not passed). Scores rank; they never assert a category."}}