Annular Khovanov homology and augmented links

Abstract

Given an annular link $L$, there is a corresponding augmented link $\widetilde{L}$ in $S^3$ obtained by adding a meridian unknot component to $L$. In this paper, we construct a spectral sequence with the second page isomorphic to the annular Khovanov homology of $L$ and it converges to the reduced Khovanov homology of $\widetilde{L}$. As an application, we classify all the links with the minimal rank of annular Khovanov homology. We also give a proof that annular Khovanov homology detects unlinks.

Figures (10)

• Figure 1: An annular link and its augmentation.
• Figure 2: Two types of smoothings.
• Figure 3: The map $\alpha_*$ and $\beta_*$.
• Figure 4: The tree corresponding to $\widetilde{U_n}$.
• Figure 5: The symmetric resolution $(10)$ of $\widetilde{U_2}$.

Theorems & Definitions (19)

• Definition 1.1
• Theorem 1.2
• Corollary 1.3
• Theorem 1.5
• Corollary 1.6
• Proposition 2.1: AP04
• Theorem 2.2: Asaeda2004CategorificationOT
• Proposition 3.1
• proof
• Theorem 3.2