论文标题
在3空间中直线纠缠的离散几何形状和拓扑
Discrete geometry and topology of entanglement of straight lines in 3-space
论文作者
论文摘要
我们提出了一个意想不到的转折，以描述n个直线的几何形状和拓扑结构，并被视为整个3D实体（因为线路是连接成对的，而链接数字为1/2或-1/2）并命名为n -cross。我们的理论源于我们在相互接触直圆柱的配置方面的工作，但是，与先前引入的环矩阵（该矩阵控制每条线的封闭方式）一起，我们现在引入了基本方向3D矩阵（其条目0、1、1和-1是线方向方向矢量的混合产物的符号）。在环和方向矩阵的空间中建立的离散运动/连接组合原理（在Loyd 15-Puzzle游戏或Khovanov同源性中形成类似于群体的矩阵并类似于移动）允许一个人可以通过基本方法辨别线条上的拓扑结构不同，而没有结的链接图。但是，在所谓的投影3D矩阵的帮助下，我们还将矩阵方法集成到结理论中，并在两种方法中建立了线路纠缠的拓扑不变性，从而将2D投影与3D配置联系起来。使用琼斯多项式，我们表明N-跨在拓扑意义上是成对连接的n个un带的链接。复制了6和7线的刚性同位素的结理论的已知结果，并给出了8行的新结果。通过我们的方法，我们达到了以前从未研究过的线条几何形状的细微差别。它可能会在代数，离散的几何学和拓扑以及量子物理学中找到应用。
We propose an unexpected twist to description of the geometry and topology of configurations of n straight lines considered as a whole 3D entity (because the lines are inseparably linked pairwise while having linking numbers 1/2 or -1/2) and named n-cross. Our theory stems from our work on configurations of mutually touching straight cylinders but, along with the previously introduced Ring matrix (that controls the encaging of each line by other lines), we now introduce fundamental direction 3D matrices (whose entries 0, 1, and -1 are signs of mixed products of line orientation vector triples). Discrete motion/connection combination principle established in the space of Ring and direction matrices (forming a groupoid and resembling moves in Loyd 15-puzzle game or Khovanov homology) allows one to discern topologically different configurations of lines with elementary methods and without link diagrams of knot theory. However, with the help of so-called projection 3D matrix we also integrated our matrix approach into the knot theory and established topological invariants for line entanglement in both approaches thus connecting 2D projections with 3D configurations. With Jones polynomials we show that an n-cross is a link of pairwise connected n unknots in a topological sense. The known results of the knot theory for rigid isotopy of 6 and 7 lines are reproduced and a novel result for 8 lines is given. With our approach we reach nuances of the geometry of lines never investigated before. It may find applications in Algebra, Discrete Geometry and Topology, and Quantum Physics.
