  Knotoids are open ended knot diagrams regarded up to Reidemeister moves and
isotopies. The notion is introduced by V.~Turaev in 2012. Two most important
numeric characteristics of a knotoid are the crossing number and the height.
The latter is the least number of intersections between a diagram and an arc
connecting its endpoints, where the minimum is taken over all representative
diagrams and all such an arcs disjoint from crossings. In the paper we answer
the question: are there any relations between the crossing number and the
height of a knotoid. We prove that the crossing number of a knotoid is greater
than or equal to twice the height of the knotoid. Combining the inequality with
known lower bounds of the height we obtain a lower bounds of the crossing
number of a knotoid via the extended bracket polynomial, the affine index
polynomial and the arrow polynomial of the knotoid. As an application of our
result we prove an upper bound for the length of a bridge in a minimal diagram
of a classical knot: the number of crossings in a minimal diagram of a knot is
greater than or equal to three times the length of a longest bridge in the
diagram.
