There are infinitely many Lissajous knots
There are infinitely many Lissajous knots
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李萨如结有无数个
DOI：
10.1007/bf02677455
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发表时间：
1997
影响因子：
0.6
通讯作者：
C. Lamm
中科院分区：
文献类型：
--
作者：
C. Lamm
Lissajous纽结由Bogle/Hearst/Jones/Stoilov在文献[1]中定义为在坐标平面上以Lissajous图形为投影的纽结。他们证明了(A)：李萨如纽结的Arf不变量一定是不变的。后来，V.Jones和J.Przytycki在[6]中引入了台球结的概念：(凸)多面体内的曲线，它是台球的轨迹。他们证明了(B)：李萨如纽结与立方台球纽结相同，以及(C)：李萨如纽结的亚历山大多项式是模2平方(这意味着较弱的结果(A))。本文作者独立地发现了(B)和(C)-见下面的定理2.3，定理3.1。作为推论，我们证明了(D)：具有两个桥的非平凡纤维结、具有奇亏格的纤维结和任意结的(p，q)-缆线不是李萨如结；特别地，非平凡的代数纽结不是李萨如结。此外，我们还证明了(E)：存在无穷多个李萨如纽结(见推论5.3)。这回答了Jones/Przytycki在[6]中提出的一个问题。
Lissajous knots have been defined by Bogle/Hearst/Jones/Stoilov in [1] as knots having Lissajous figures as their projections on the coordinate planes. They proved (A): the Arf invariant of a Lissajous knot must vaaish. Later V. Jones and J. Przytycki introduced in [6] the concept of billiard knots: curves inside a (convex) polyhedron which are the trajectory of a billiard ball. They showed (B): Lissajous knots are the same as billiard knots in a cube, and (C): the Alexander polynomial of a Lissajous knot is a square modulo 2 (which implies the weaker result (A)).The present author has independently found (B) and (C)--see Theorem 2.3, Theorem 3.1 below. As a corollary we show (D): non-trivial fibred knots with two bridges, fibred knots with odd genus and (p, q)-cables (with] p],] q]> 1) of arbitrary knots are not Lissajous knots; in particular, non-trivial algebraic knots are not Lissajous. In addition we prove (E): there are infinitely many Lissajous knots (see Corollary 5.3). This answers a question of Jones/Przytycki in [6].