Annular Khovanov homology detects three-strand weaving links

Abstract

For $N\geq1$, let $K_N$ be the annular closure of $(\sigma_1\sigma_2^{-1})^N.$ We prove that triply graded annular Khovanov homology over $\mathbb F_2$ detects the underlying unoriented annular link $K_N$. If $3\nmid N$, the only ambiguity is overall orientation reversal. If $3\mid N$, the only ambiguity is independent reversal of components; every such reorientation has the same homology, so this is sharp. The proof combines braid detection from the extremal annular grading with a rigidity theorem: the Jones polynomial and exponent sum determine $(\sigma_1\sigma_2^{-1})^N$ up to conjugacy in $B_3$.