Knots in $\mathbb{R}P^3$

Abstract

This paper studies knots in three dimensional projective space. Our technique is to associate a virtual link to a link in projective space so that equivalent projective links go to equivalent virtual links (modulo a special flype move). We apply techniques in virtual knot theory to obtain a Jones polynomial for projective links. We show that this is equivalent to the known Jones polynomial defined by Drobotukhina for them. We apply virtual Khovanov homology and the virtual Rasmussen invariant of Dye, Kaestner, and Kauffman to projective links. We compare this cohomology theory with the Khovanov type theory developed by Manolescu and Willis for projective knots. We show that these theories are essentially equivalent.