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

For a non-split multi-crossing diagram D of a link L we show that [Formula: see text] holds. Here, [Formula: see text] is the arc index and [Formula: see text] 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 weak rectangular diagram can be converted to usual rectangular diagram preserving its arc index.