论文标题
表面和音量界面上的交替链接
Alternating links on surfaces and volume bounds
论文作者
论文摘要
弱广泛的交替结是打结，在紧凑的3个manifold中的交替投影对封闭的表面上的交替投影，它们具有许多通常的交替结的双曲线几何特性。例如，通常的交替结的体积在上方和下方是由于拉力比的扭曲数。 Howie和Purcell表明，对于弱广泛的交替结，类似的下限也具有相似的下限。在本文中，我们表明，在三个球体中产生一个弱概括的交替结的家族，并表明了固定的扭曲数量，但没有固定的扭曲数量，但无限的体积是通过产生一个弱概括的交替结的家族。与通常的交替结相比，作为推论，通用的交替结可能具有任意小的尖缘密度，其尖连孔密度由于拉力和purcell而被偏离零。另一方面，我们证明了弱概括的交替投影的扭曲数确实在增厚表面内的体积上具有两个方面的线性边界。我们陈述了一些相关的开放问题。
Weakly generalised alternating knots are knots with an alternating projection onto a closed surface in a compact irreducible 3-manifold, and they share many hyperbolic geometric properties with usual alternating knots. For example, usual alternating knots have volume bounded above and below by the twist number of the alternating diagram due to Lackenby. Howie and Purcell showed that a similar lower bound holds for weakly generalised alternating knots. In this paper, we show that a generalisation of the upper volume bound does not hold, by producing a family of weakly generalised alternating knots in the 3-sphere with fixed twist number but unbounded volumes. As a corollary, generalised alternating knots can have arbitrarily small cusp density, in contrast with usual alternating knots whose cusp densities are bounded away from zero due to Lackenby and Purcell. On the other hand, we show that the twist number of a weakly generalised alternating projection does gives two sided linear bounds on volume inside a thickened surface; we state some related open questions.
