Quantum Topology Seminar
Rama Mishra
IISER, Pune
Knots, Links and spherical braids in $\mathbb{R}P^3$
Abstract: (Rama is visiting at UIC. The talk will be broadcast from 530SEO and Rama will be available for discussion in 530 SEO after the talk.)
We study some geometric features of knots in real projective 3-space, $\mathbb{R}P^3$. Since knots in $\mathbb{R}P^3$ are classified into three disjoint classes: {\it affine, class-0 non-affine and class-1} knots, it is natural to wonder in which class a given knot belongs to. We provide a structure theorem for these knots which helps in studying this classification.
We propose a procedure called `{\it space bending surgery }' on affine knots to produce several examples of non affine knots in $\mathbb{R}P^3$. We also define a notion of the {\it genus } for knots in $\mathbb{R}P^3$ and study some of its properties. We introduce the notion of {\it companionship} of knots in $\mathbb{R}P^3$ and using that we provide a geometric criteria for a knot to be affine.
Furthermore we develop a braid theory in real projective 3 space, by generalizing the classical plat closures of spherical braids. We prove that every link in real projective 3 space is isotopic to a plat closure of some spherical braid. We introduce some moves on spherical braids whose plat closure are isotopic as links. We hope to prove a theorem analogous to the {\it Markov theorem} in classical knot theory. This theorem may be used to construct quantum invariants for knots in real projective 3 space in future.
Thursday May 18, 2023 at 12:00 PM in Zoom