Weak rectangular diagrams, multi-crossing number, and arc index

2025/07/02 by Tetsuya Ito, Ito, Tetsuya
Computer Science · #Data Management and Algorithms #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.2507.01404

openalex publication_date 2025/07/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For a non-split multi-crossing diagram D of a link L we show that α(L)-2 ≤ c₂(D) + ∑_{n> 2}(2n-4)c_n(D) holds. Here α(L) is the arc index and c_n(D) is the number of n-crossings of D. This generalizes and subsumes many known inequalities related to multi-crossing numbers. In the course of proof, we introduce a notion of weak rectangular diagram and show that a loose rectangular diagram can be converted to usual rectangular diagram preserving its arc index.

Citations

• Bisected vertex leveling of plane graphs: braid index, arc index and delta diagrams
• Triple Crossing Number and Double Crossing Braid Index
• Knot projections with a single multi-crossing
• Triple Crossing Number of Knots and Links

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• Bisected vertex leveling of plane graphs: braid index, arc index and delta diagrams