Annular Khovanov homology detects three-strand weaving links

Abstract

For $N\ge 1$, 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 $B_3$.

9 pages