论文标题
表面上交替结的尖端体积
Cusp Volumes of Alternating Knots on Surfaces
论文作者
论文摘要
我们研究双曲线结的几何形状，这些结的几何形状可以接收封闭的3个序列中嵌入式表面上的交替投影。我们表明，在温和的假设下，它们的尖端区域在交替投影的扭曲数和投影表面的属方面接受了两个方面的边界。结果，我们得出了斜率长度的示意估计值，并为Dehn手术提供了应用。这些概括性的结果是3球中有关交替结的Lackenby和Purcell的结果。 使用Kalfagianni和Purcell的结果，我们指出，在较高属的表面上的交替结可以任意尖端密度很小，而与球体上的交替结相比，其尖密度由于缺乏和Purcell而偏离零。
We study the geometry of hyperbolic knots that admit alternating projections on embedded surfaces in closed 3-manifolds. We show that, under mild hypothesis, their cusp area admits two sided bounds in terms of the twist number of the alternating projection and the genus of the projection surface. As a result, we derive diagrammatic estimates of slope lengths and give applications to Dehn surgery. These generalize results of Lackenby and Purcell about alternating knots in the 3-sphere. Using a result of Kalfagianni and Purcell, we point out that alternating knots on surfaces of higher genus, can have arbitrarily small cusp density, in contrast to alternating knots on spheres whose cusp densities are bounded away from zero due to Lackenby and Purcell.
