论文标题
交叉数与结的高度之间的关系
A relation between the crossing number and the height of a knotoid
论文作者
论文摘要
打结是开放式结节图，该图被认为是雷德氏移动和同位素。该概念是由V.〜turaev在2012年引入的。一个结节的两个最重要的数字特征是交叉数和高度。后者是一个图和连接其端点的弧之间的相交数量最少，其中最小值是在所有代表性图上占据的最小值，以及所有这些ARCS与交叉点的不相交。在论文中，我们回答了一个问题：交叉数与缝合的高度之间有任何关系。我们证明，缝合的交叉数大于或等于结节高度的两倍。将不平等与高度的已知下限结合在一起，我们通过扩展的支架多项式，仿射指数多项式和弦的箭头多项式获得了一个结的交叉数的下限。作为结果的应用，我们证明了在经典结的最小图中的桥梁长度的上限：结中的交叉数量大于或等于图中最长桥的三倍。
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.
