论文标题
在不可取向的4助双扭结上
On the nonorientable 4-genus of double twist knots
论文作者
论文摘要
我们研究了一个由2桥结，扭结和双重扭结$ c（m，n）$的特殊家族的不可取向的4摄氏度$γ_4$。由于不可取向的4摄氏度是由不可取向的3天界定的，因此众所周知，$γ_4（c（m，n））\ le 3 $。通过使用显式结构来获得$γ_4$的上限，并使用唐纳森的对角线定理得出的已知障碍物以获得$γ_4$的下限，我们产生了$ c（m，n）$ $ c（m，n）$γ_4= 0,1,1,2，$和$ 3 $的无限亚科。但是，仍然存在无限的双重扭结，其中我们的作品仅表明$γ_4$位于$ \ {1,2 \}，\ {2,3 \} $中的一组中，或$ \ {1,2,3 \} $。我们将所有$ C（m，n）$的结果与$ | M | $和$ | n | $最高50。我们还提供了无限数量的示例，这些示例回答了穆拉卡米和Yasuhara的猜想。
We investigate the nonorientable 4-genus $γ_4$ of a special family of 2-bridge knots, the twist knots and double twist knots $C(m,n)$. Because the nonorientable 4-genus is bounded by the nonorientable 3-genus, it is known that $γ_4(C(m,n)) \le 3$. By using explicit constructions to obtain upper bounds on $γ_4$ and known obstructions derived from Donaldson's diagonalization theorem to obtain lower bounds on $γ_4$, we produce infinite subfamilies of $C(m,n)$ where $γ_4=0,1,2,$ and $3$, respectively. However, there remain infinitely many double twist knots where our work only shows that $γ_4$ lies in one of the sets $\{1,2\}, \{2,3\}$, or $\{1,2,3\}$. We tabulate our results for all $C(m,n)$ with $|m|$ and $|n|$ up to 50. We also provide an infinite number of examples which answer a conjecture of Murakami and Yasuhara.
