Knots, Links and Spherical Braids in Real Projective 3-space

Abstract

This thesis studies some geometric properties of knots in real projective 3-space. 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. 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. Later it is shown that this operation can be altered as a {\it class changing surgery} on an arbitrary knot in $\mathbb{R}P^3$. We then study the notion of companionship of knots in $\mathbb{R}P^3$ and provide a geometric criterion for a knot to be affine. We also define an invariant called \say{genus} for knots in $\mathbb{R}P^3$ and show that this genus detects knottedness and gives a criterion for a knot to be affine and of class-$1$. Using the properties of genus we prove a \say{non-cancellation} theorem for both the surgery operations mentioned above. We introduce a notion of {\it{cable knot} } and show that cable knots with a class $1$ companion knot completely characterize genus $1$ knots.

In the later part of the thesis, we develop a method for constructing links in $\mathbb{R}P^3$ as plat closures of spherical braids. This method is a generalization of the concept of \say{plats} in $S^3$ defined by Joan Birman \cite{Bir}. We prove that any link in $\mathbb{R}P^3$ can be constructed in this manner. We introduce a new kind of permutation (called \textit{residual permutation}) associated with a spherical braid in $\mathbb{R}P^3$ and prove that the number of disjoint cycles in the residual permutation of a spherical braid is equal to the number of components of the plat closure link of this braid. This braid representation provides another criterion for detecting affineness. We develop a set of moves on spherical braids in the same spirit as the classical Markov moves on braids. Braids related by a finite sequence of these moves will produce isotopic plat closure links. We conjecture that the converse of this is also true, so we hope to have an analogue of Markov theorem for links in $\mathbb{R}P^3$.