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

TL;DR

The paper studies knots in $\mathbb{R}P^3$ by transferring them to virtual links via a Flype-aware model, enabling a Jones-type polynomial from the virtual bracket that matches Drobotukhina's invariant. It extends Khovanov and Rasmussen-type invariants to projective knots through the map $\pi$ to virtual knots, and shows these invariants align with Manolescu–Willis theory, including cobordism and genus results for positive knots. It provides a comprehensive framework linking virtual knot theory to projective knot theory, along with a comparison of cohomology theories and explicit limitations where distinct projective knots share the same virtual image. The work thus offers a unified perspective and practical tools for studying projective knots while highlighting intrinsic blind spots and open problems to pursue further.

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.

Figures (38)

• Figure 1: Slide Moves
• Figure 2: Sliding a crossing over a half twist
• Figure 3: Association of a virtual link to a projective link.
• Figure 4: Defining rules for the bracket polynomial.
• Figure 5: Flype move

Theorems & Definitions (19)

• Proposition 2.1
• proof
• Corollary 2.2
• Corollary 2.3
• Remark 2.1
• Theorem 2.4
• proof
• Theorem 2.5
• proof
• Corollary 2.6