论文标题
双重扭结的左订购手术ii
Left orderable surgeries of double twist knots II
论文作者
论文摘要
如果$ r $ surgery沿着$ k $获得的3个manifold沿$ k $获得的3个manifold，则称为打结$ k \ subset s^3 $的左订购斜率。考虑两桥结$ c（200万，\ pm 2n）$和$ c（200万+1，-2n）$在conway符号中，其中$ m \ ge 1 $和$ n \ ge 2 $是整数。通过使用\ textIt {连续}夸张$ \ mathrm {sl} _2（\ mathbb {r}）$ - 结组的表示，在\ cite {ht-genus1，tr}中显示了任何slope in $ in $（4n，4n，4n，4n，4m）$（4n，4m）$ [0，$ ax。 slope of $C(2m, 2n)$ (resp. $C(2m, - 2n)$) and in \cite{Ga} that any slope in $(-4n,0]$ is a left orderable slope of $C(2m+1,-2n)$. However, the proofs of these results are incomplete since the \textit{continuity} of the families of representations was not proved. In this paper, we complete these此外，我们证明了$（-4N，4M）$中的任何坡度都是$ C（2M+1，-2n）$的左订购坡度，由双曲线$ \ Mathrm {sl} _2（\ Mathbb {r}）$ - neot组的表示。
A slope $r$ is called a left orderable slope of a knot $K \subset S^3$ if the 3-manifold obtained by $r$-surgery along $K$ has left orderable fundamental group. Consider two-bridge knots $C(2m, \pm 2n)$ and $C(2m+1, -2n)$ in the Conway notation, where $m \ge 1$ and $n \ge 2$ are integers. By using \textit{continuous} families of hyperbolic $\mathrm{SL}_2(\mathbb{R})$-representations of knot groups, it was shown in \cite{HT-genus1, Tr} that any slope in $(-4n, 4m)$ (resp. $[0, \max\{4m, 4n\})$) is a left orderable slope of $C(2m, 2n)$ (resp. $C(2m, - 2n)$) and in \cite{Ga} that any slope in $(-4n,0]$ is a left orderable slope of $C(2m+1,-2n)$. However, the proofs of these results are incomplete since the \textit{continuity} of the families of representations was not proved. In this paper, we complete these proofs and moreover we show that any slope in $(-4n, 4m)$ is a left orderable slope of $C(2m+1,-2n)$ detected by hyperbolic $\mathrm{SL}_2(\mathbb{R})$-representations of the knot group.
