论文标题
平面浸入图中的循环的交叉数量和旋转数
Crossing numbers and rotation numbers of cycles in a plane immersed graph
论文作者
论文摘要
对于将彼得森图浸入平面中的任何通用浸入平面中，距离两个边缘之间的交叉点数是奇怪的。所有$ 5 $ - 周期的交叉数量的总和很奇怪。所有$ 5 $ -CYCLE的旋转数的总和是偶数。我们以$ 6 $ -CYCLES，$ 8 $ -CYCLES和$ 9 $ -CYCLES的价格显示出类似的结果。对于彼得森图的任何legendrian空间嵌入，都有一个$ 5 $循环，这不是一个没有最大的瑟斯顿·奔驰编号的毫无管的，而且所有瑟斯顿·奔驰数量的周期数的总和是$ 7 $ $ 7 $ $ $ 7 $ $ $ $ $ $ $ $ $ $倍的瑟斯顿·贝尼奎林数量的总和。我们显示出Heawood图的类似结果。我们还为某些图显示了其他一些结果。我们表征了抽象图，该图形将通用浸入其所有周期都旋转数量$ 0 $的平面中。
For any generic immersion of a Petersen graph into a plane, the number of crossing points between two edges of distance one is odd. The sum of the crossing numbers of all $5$-cycles is odd. The sum of the rotation numbers of all $5$-cycles is even. We show analogous results for $6$-cycles, $8$-cycles and $9$-cycles. For any Legendrian spatial embedding of a Petersen graph, there exists a $5$-cycle that is not an unknot with maximal Thurston-Bennequin number, and the sum of all Thurston-Bennequin numbers of the cycles is $7$ times the sum of all Thurston-Bennequin numbers of the $5$-cycles. We show analogous results for a Heawood graph. We also show some other results for some graphs. We characterize abstract graphs that has a generic immersion into a plane whose all cycles have rotation number $0$.
