Intrinsic Khovanov homology in $\mathbb{RP}^3$

TL;DR

This work extends Khovanov-type homology to links in unparametrized $\mathbb{RP}^3$ by embedding the theory in Bar-Natan’s categorical framework on $\mathbb{RP}^2$ and distinguishing class-0 and class-1 links via appropriate coefficient choices and target categories. A central technical tool is a filtration that reduces complex computations to crossingless diagrams, together with a detailed analysis of the sweep-around move, which is essential to upgrading the invariant to unparametrized $\mathbb{RP}^3$ and establishing functoriality for link cobordisms in $I\times\mathbb{RP}^3$. The paper proves that the resulting Khovanov homology is an invariant of links in unparametrized $\mathbb{RP}^3$ and presents two intrinsic formulations: a class-1 theory over $\mathbf{BN}_{\mathbb{RP}^2,1}$ and a class-0 theory over a quotient $\mathbf{BN}_{\mathbb{RP}^2,0}'$, with the former recovering Bar-Natan’s bracket on $\mathbb{RP}^3$. These results connect to broader efforts to generalize Khovanov homology beyond $S^3$ and illuminate interactions with the mapping class group of $\mathbb{RP}^3$ and related Floer-theoretic approaches.

Abstract

We prove that Khovanov homology is an invariant of links in unparametrized $\mathbb{RP}^3$'s, i.e., oriented $3$-manifolds diffeomorphic to $\mathbb{RP}^3$. Along the way, we establish the functoriality of Khovanov homology for link cobordisms in $I\times\mathbb{RP}^3$.

Figures (10)

• Figure 1: A two-component class-1 link in $I\tilde{\times}\mathbb{RP}^2$ drawn in the cylinder model (left), and its diagram on $\mathbb{RP}^2$ drawn in the disk model (right). On the left, the ends of the strands are on the boundary of the cylinder.
• Figure 2: The dot relation in $\mathbf{BN}_{\mathbb{RP}^2}$.
• Figure 3: The neck-cutting relation in $\mathbf{BN}_{\mathbb{RP}^2}$.
• Figure 4: The sweep-around move in $I\tilde{\times}\mathbb{RP}^2$ shown in the cylinder model. The first three maps isotope the unknot component near the whole boundary. The fourth map isotopes the unknot component near the upper boundary. The last map flips the unknot component locally.
• Figure 5: The sweep-around move in $I\tilde{\times}\mathbb{RP}^2$ in diagram shown in the disk model.

Theorems & Definitions (32)

• Theorem 1.1
• Theorem 1.2
• Remark 1.3
• Example 2.1
• Remark 2.2
• Theorem 3.1
• Theorem 3.2
• Remark 3.3
• Remark 3.4
• Proposition 3.5