Geometry of knots in real projective 3-space

Abstract

This paper discusses some geometric ideas associated with knots in real projective 3-space Double-struck capital RP3. These ideas are borrowed from classical knot theory. Since knots in Double-struck capital RP3 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 behavior near the projective plane at infinity. We propose a procedure called 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 Double-struck capital RP3. We then study the notion of companionship of knots in Double-struck capital RP3 and using it we provide geometric criteria for a knot to be affine. We also define a notion of "genus" for knots in Double-struck capital RP3 and study some of its properties. We prove that this genus detects knottedness in Double-struck capital RP3 and gives some criteria for a knot to be affine and of class-1. We also prove a "non-cancellation" theorem for space bending surgery using the properties of genus. Then we show that a knot can have genus 1 if and only if it is a cable knot with a class-1 companion. We produce examples of class-0 non-affine knots with genus 1. Thus we highlight that, Double-struck capital RP3 admits a knot theory with a truly different flavor than that of S3 or Double-struck capital R3.