论文标题
真正投影的结几何形状$ 3 $--太空
Geometry of knots in real projective $3$-space
论文作者
论文摘要
本文讨论了一些与打结相关的几何思想3 $ \ mathbb {r} p^3 $。这些想法是从古典结理论中借来的。由于$ \ mathbb {r} p^3 $中的结成三个不相交类，因此 - 仿射，班级 - $ 0 $ non-affine and class- $ 1 $节，因此自然而然地想知道A级给定的结属于哪个级别。在本文中，我们试图回答这个问题。我们提供 这些结的结构定理，有助于描述其在无穷大的投射平面附近的行为。 我们提出了一个称为{\ IT空间弯曲手术}的程序，以产生几个结的示例。后来，我们证明可以在$ \ mathbb {r} p^3 $中以任意结上扩展此操作。我们还为$ \ mathbb {r} p^3 $中的结的\ Say {属}定义了一个概念，并研究了其一些属性。我们证明，该属在$ \ mathbb {r} p^3 $中检测到打结，并给出了一些结构标准，即结成仿射和班级 - $ 1 $。我们还证明了使用属的特性进行空间弯曲手术的{non-Cancellation}定理。我们制作了班级的例子 - $ 0 $ $ 1 $ $ 1 $。最后，我们研究了$ \ mathbb {r} p^3 $中结的陪伴的概念，并使用它为结的几何标准提供了仿射。 因此，我们强调，$ \ mathbb {r} p^3 $接受了与$ s^3 $或$ \ MATHBB {r}^3 $的结论。
This paper discusses some geometric ideas associated with knots in real projective 3-space $\mathbb{R}P^3$. These ideas are borrowed from classical knot theory. Since knots in $\mathbb{R}P^3$ are classified into three disjoint classes, - affine, class-$0$ non-affine and class-$1$ knots, it is natural to wonder in which class a given knot belongs to. In this paper we attempt to answer this question. We provide a structure theorem for these knots which helps in describing their behaviour near the projective plane at infinity. We propose a procedure called {\it space bending surgery}, on affine knots to produce several examples of knots. We later show that this operation can be extended on an arbitrary knot in $\mathbb{R}P^3$. We also define a notion of \say{ genus} for knots in $\mathbb{R}P^3$ and study some of its properties. We prove that this genus detects knottedness in $\mathbb{R}P^3$ and gives some criteria for a knot to be affine and of class-$1$. We also prove a \say{non-cancellation} theorem for space bending surgery using the properties of genus. We produce examples of class-$0 $ non-affine knots with genus $1$. And finally we study the notion of companionship of knots in $\mathbb{R}P^3$ and using that we provide a geometric criteria for a knot to be affine. Thus we highlight that, $\mathbb{R}P^3$ admits a knot theory with a truly different flavour than that of $S^3$ or $\mathbb{R}^3$.
