论文标题
奇数多个结的新界限和链接
A New Bound on Odd Multicrossing Numbers of Knots and Links
论文作者
论文摘要
$ n $ link $ l $的$ n $交叉投影是在飞机上的$ l $投影，以至于$ l $上的$ n $点在每个交叉口都彼此叠加。我们证明了所有$ k \ in \ mathbb {n} $和所有链接$ l $，不等式$$ c_ {2k+1}（l）\ geq \ frac \ frac {2g（l）+r（l）+r（l）+r（l）-1} -1} {k^2} {k^2} $$ $（2K+1）$ - 交叉数字，$ 3 $ - 属和$ L $的组件数。该结果用于证明在圆环结的奇数交叉数上是一个新的结合，并概括了Jablonowski的结果。 我们还证明了在$ 5 $划分的2螺道结和链接上的新上限。此外，我们改善了$ 5 $划分的$ 79 $节的较低界限，并以$ 2 $ - 跨的数字$ \ leq 12 $。最后，我们改善了$ 5 $节的$ 7 $加入数字的下限，并以$ 2 $ - 串在一起的数字$ \ leq 12 $。
An $n$-crossing projection of a link $L$ is a projection of $L$ onto a plane such that $n$ points on $L$ are superimposed on top of each other at every crossing. We prove that for all $k \in \mathbb{N}$ and all links $L$, the inequality $$c_{2k+1}(L) \geq \frac{2g(L) + r(L)-1}{k^2}$$ holds, where $c_{2k+1}(L)$, $g(L)$, and $r(L)$ are the $(2k+1)$-crossing number, $3$-genus, and number of components of $L$ respectively. This result is used to prove a new bound on the odd crossing numbers of torus knots and generalizes a result of Jablonowski. We also prove a new upper bound on the $5$-crossing numbers of the 2-torus knots and links. Furthermore, we improve the lower bounds on the $5$-crossing numbers of $79$ knots with $2$-crossing number $ \leq 12$. Finally, we improve the lower bounds on the $7$-crossing numbers of $5$ knots with $2$-crossing number $\leq 12$.
