The semimeander crossing number of knots and related invariants

The minimum number of crossings among all of the diagrams of a knot $K$ composed of at most $k$ smooth simple arcs is called the $k$-arc crossing number of $K$. This number is denoted by $\mathrm{cr}_k(K)$. The $2$-arc crossing number is also called the semimeander crossing number. The article studies connections of the $k$-arc crossing numbers with the classical crossing number $\mathrm{cr}(K)$ of $K$. It is proved that for each knot $K$, the following inequalities are fulfilled: $\mathrm{cr}_2(K) \leqslant \sqrt[4]{6}^{\mathrm{cr}(K)}$ and $\mathrm{cr}_k(K) \leqslant \mathrm{cr}_{k+1}(K) + \frac{(\mathrm{cr}_{k+1}(K))^2} {2(k+1)^2}$.