The region index and the unknotting number of a knot
The region index and the unknotting number of a knot
复制标题
结点的区域索引和解结数
DOI：
10.1016/j.topol.2017.01.012
复制
发表时间：
2017
影响因子：
0.6
通讯作者：
Toshifumi Tanaka
中科院分区：
文献类型：
--
作者：
T. Kato;Toshifumi Tanaka
对于3-球面S 3中的任何节点，都存在一个图，使得如果改变图中某一区域边界上的所有交点，我们就有这个未知节点。所有这类图的交叉变化的最小数目称为结点的区域索引。显然，解结数小于或等于区域指数。利用Goeritz不变量证明了任意正整数m(m≥2)的解结数m与区域指数之间存在间隙的纽结.我们还利用拉斯穆森不变量证明了对于任意正整数n(n≥2)，存在一个解结数和区域指数等于n的纽结。
For any knot in the 3-sphere S 3, there exists a diagram such that we have the unknot if we change all crossings on the boundary of some region of the diagram. The minimal number of the crossing changes over all such diagrams is called the region index of a knot. Clearly, the unknotting number is less than or equal to the region index. In this paper, we show that there exists a knot which has a gap between the unknotting number m and the region index for any positive integer m (m≥ 2) by using the Goeritz invariant. We also show that there exists a knot which has the unknotting number and the region index that are equal to n for any positive integer n (n≥ 2) by using the Rasmussen invariant.