Knotoids are open ended knot diagrams regarded up to Reidemeister moves and isotopies. The notion is introduced by 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 arcs which are 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.

中文翻译：

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交叉数与结点高度之间的关系

Knotoids 是与 Reidemeister 移动和同位素相关的开放式结图。这个概念是 Turaev 在 2012 年提出的。结节的两个最重要的数字特征是交叉数和高度。后者是图表和连接其端点的弧之间的交叉点数量最少，其中最小值适用于所有代表性图表以及与交叉点不相交的所有此类弧。在论文中，我们回答了这个问题：交叉数和结点的高度之间是否存在任何关系。我们证明了结点的交叉数大于或等于结点高度的两倍。将不等式与已知的高度下界相结合，我们通过扩展括号多项式获得结点交叉数的下界，仿射指数多项式和节点的箭头多项式。作为我们结果的应用，我们证明了经典结的最小图中桥梁长度的上限：结的最小图中的交叉数大于或等于最长结的长度的三倍图中的桥。