论文标题
两桥结组的横帽数和表达
Crosscap number and epimorphisms of two-bridge knot groups
论文作者
论文摘要
我们考虑了所有原始结的横盖数$γ$与部分订单之间的关系，这定义如下。对于两个节$ k $和$ j $，我们说$ k \ geq j $如果存在表达$ f：π_1（s^3-k）\ longrightArrowπ_1（s^3-j）$。我们证明，如果$ k $和$ j $是2桥结和$ k> j $，则$γ（k）\ geq3γ（j）-4 $。我们还对不平等的所有对$（k，j）$进行了分类。铃木和Tran证明了有关两个结属的类似结果。也就是说，如果$ k $和$ j $是2桥结和$ k> j $，则$ g（k）\ geq 3 g（j）-1 $，其中$ g（k）$表示结$ k $的属。
We consider the relationship between the crosscap number $γ$ of knots and a partial order on the set of all prime knots, which is defined as follows. For two knots $K$ and $J$, we say $K \geq J$ if there exists an epimorphism $f:π_1(S^3-K) \longrightarrow π_1(S^3-J)$. We prove that if $K$ and $J$ are 2-bridge knots and $K> J$, then $γ(K) \geq 3γ(J) -4$. We also classify all pairs $(K,J)$ for which the inequality is sharp. A similar result relating the genera of two knots has been proven by Suzuki and Tran. Namely, if $K$ and $J$ are 2-bridge knots and $K >J$, then $g(K) \geq 3 g(J)-1$, where $g(K)$ denotes the genus of the knot $K$.
