AI 中文总结
研究\(K_N\)为\((\sigma_1\sigma_2^{-1})^N\)环形闭包，证明\(\mathbb F_2\)上三重分次环形霍万诺夫同调可检测其底层无向环形链，通过结合辫子检测与刚性定理，明确不同\(N\)值下链的方向与同调关系。
AI 中文摘要
对于\(N\geq1\)，令\(K_N\)为\((\sigma_1\sigma_2^{-1})^N\)的环形闭包。我们证明了在\(\mathbb F_2\)上的三重分次环形霍万诺夫同调能检测出底层的无向环形链\(K_N\)。若\(3\nmid N\)，唯一的模糊性是整体方向反转；若\(3\mid N\)，唯一的模糊性是各分量独立反转，且每种这样的重新定向具有相同的同调，这是精确的。证明结合了来自极值环形分次的辫子检测和一个刚性定理：琼斯多项式和指数和在\(B_3\)中确定\((\sigma_1\sigma_2^{-1})^N\)到共轭。
英文摘要
For $N\geq1$, let $K_N$ be the annular closure of $(σ_1σ_2^{-1})^N.$ 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 $(σ_1σ_2^{-1})^N$ up to conjugacy in $B_3$.
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