On computational complexity of Khovanov homology

TL;DR

This work investigates the computational complexity of Khovanov homology, showing that Kh can be computed in polynomial time for closed $3$-braids while Bar-Natan’s scanning remains exponential on certain simple $3$-braids. It introduces extremal-degree Kh algorithms $\mathcal{A}_k$ (and $\mathcal{B}_{k,t}$) based on nice scanning sequences, enabling polynomial-time recovery of Kh in extreme homological degrees and establishing binomial-rank upper bounds. The authors also prove structural Kh decompositions for certain $3$-braid families (Omega cases) and derive asymptotic lower bounds on rank, illustrating the limits of the bounds and the potential for optimization. Together, these results illuminate when Kh is tractable and provide concrete algorithms and bounds that guide computation and complexity understanding in knot homology, with open questions guiding future work on higher strands and torus links.

Abstract

Computing the Jones polynomial of general link diagrams is known to be $\#$P-hard, while restricting the computation to braid closures on fixed number of strands allows for a polynomial time algorithm. We investigate polynomial time algorithms for Khovanov homology of braids and show that for $3$-braids there is one. In contrast, we show that Bar-Natan's scanning algorithm runs in exponential time when restricted to simple classes of $3$-braids. For more general braids, we obtain that a variation of the scanning algorithm computes the Khovanov homology for a bounded set of homological degrees in polynomial time. We also prove upper and lower bounds on the ranks of Khovanov homology groups.

Figures (9)

• Figure 1: The dotted relations of $\mathcal{C}ob_{\bullet/l}^{\mathbb Z}(B,\dot{B})$. We refer to the two leftmost as sphere relations, the middle one as double dot relation and the right one as neck cutting relation.
• Figure 2: The local delooping isomorphism $\mathfrak d_1\mathfrak d_2$ its inverse $\mathfrak r_1,\mathfrak r_2$.
• Figure 3: The tangles for $\sigma_i$, $\sigma^{-1}_i$, and $\sigma_1\sigma_3\sigma_2^{-1}$.
• Figure 4: The cochain complex $\mathcal{B}_k$ over $\mathcal{C}ob_{\bullet/l}^\mathbb{Z}(B^2_2)$. The object with grading shift $q^{6k}$ is in homological degree $4k$, and the object with grading shift $q^{-1}$ is in homological degree $0$. The complex continues in negative homological degrees via the two objects connected by the $e$-morphism. The letter $S$ stands for a surgery between the two smoothings, and $D$ stands for back-and-forth surgeries.
• Figure 5: The cochain complex $\mathcal{C}_k$. Notice that $(2X)^i$ is $0$ for $i\geq 2$, and $1$ for $i=0$. Homological degrees are indicated by $u^i$ for some modules.

Theorems & Definitions (45)

• Conjecture 1.1: Przytycki, Silvero zbMATH07862431
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Proposition 1.5
• Proposition 1.6
• Lemma 2.1: Gaussian elimination
• Theorem 2.2: Sköldberg MR2171225
• proof : Sketch of proof
• proof : Sketch of proof