On the Khovanov homology of 3-braids

TL;DR

The paper resolves the Przytycki–Sazdanović conjecture by showing that the Khovanov homology of closures of 3-braids contains only $2$-torsion, completing the Murasugi classification analysis with Bar-Natan’s tangle framework, delooping, and the reduced BLT spectral sequence. It develops explicit Bar-Natan–type complexes for torus tangles/links and for a broad family of words in the seven Murasugi classes, obtaining precise decompositions into shifted copies of A, A(1), and A(2) that govern torsion structure. The authors prove Knight Move behavior for 3-braids and establish spectral sequence collapse results that force absence of odd torsion, providing a clear, computable description of the reduced and unreduced Khovanov homology in these cases. The results sharpen understanding of torsion phenomena in Khovanov homology, especially for low-strand braids, and offer a systematic methodological framework for analyzing similar questions via Bar-Natan’s tangle formalism and BLT-type spectral sequences.

Abstract

We prove the conjecture of Przytycki and Sazdanovic that the Khovanov homology of the closure of a 3-stranded braid only contains torsion of order 2. This conjecture has been known for six out of seven classes in the Murasugi-classification of 3-braids and we show it for the remaining class. Our proof also works for the other classes and relies on Bar-Natan's version of Khovanov homology for tangles as well as his delooping and cancellation techniques, and the reduced integral Bar-Natan--Lee--Turner spectral sequence. We also show that the Knight-move conjecture holds for 3-braids.

Figures (13)

• Figure 1: The tangles for $a$, $b$, and $abb^{-1}$.
• Figure 2: The standard smoothings and their morphisms. The letter $S$ stands for a surgery, and $D$ for two surgeries. The morphisms $c,d$, and $e$ have the same domain and codomain, which can be $\alpha,\beta,\gamma,\delta$, or $\tilde{\alpha}$ for $c,d$, and $\tilde{\omega}$ for $e$.
• Figure 3: The complex $\mathcal{B}\otimes \varepsilon$ after delooping.
• Figure 4: The complex $\mathcal{B}_{3k}\otimes \zeta$ after delooping.
• Figure 5: The complex $\mathcal{B}_{3k}\otimes \varepsilon$ after delooping.

Theorems & Definitions (59)

• Conjecture 1.1: Przytycki--Sazdanović MR3205574
• Theorem 1.2
• Conjecture 1.3: Knight Move Conjecture MR1917056
• Theorem 1.4
• Definition 2.1
• Proposition 3.1
• proof
• Definition 3.2
• Corollary 3.3
• proof