Geometry of knots in real projective $3$-space

Abstract

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 $\mathbb{R}^3$.