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

For a non-split multi-crossing diagram $D$ of a link $L$ we show that $\alpha (L)-2\le c_2(D)+{\sum }_{n>2}(2n-4)c_n(D)$ holds. Here $\alpha (L)$ 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.