AI 中文总结
本文证明了Bar-Natan不变量经$A_\infty$-扩张得到的扭曲复形是缠绕不变量，解决了Kotelskiy等人的猜想，并通过推广的矩阵多因子分解和Koszul形变收缩公式给出证明。
AI 中文摘要
Kotelskiy、Watson和Zibrowius证明了与4端有向缠绕$T$相关的Bar-Natan不变量$\unicode{x0414}(T)$可扩张为Bar-Natan代数的$A_\infty$-增强上的扭曲复形$\unicode{x0414}^\infty(T)$。我们证明$\unicode{x0414}^\infty(T)$本身是一个缠绕不变量，解决了他们工作中的猜想。我们从一个关于\emph{矩阵多因子分解}$\mathcal M(T)$的不变性陈述推导出这一点，将Ballinger的构造推广到系数环$\mathbb Z[G]$。该多因子分解在1-同伦等价的意义下是良定义的。证明依赖于Koszul矩阵因子分解的特殊形变收缩的显式公式，这些公式在此设置中实现了去环化，可能具有独立的意义。
英文摘要
Kotelskiy, Watson and Zibrowius show that the Bar-Natan invariant $\unicode{x0414}(T)$ associated to a $4$-ended oriented tangle $T$ extends to a twisted complex $\unicode{x0414}^\infty(T)$ over an $A_\infty$-enhancement of the Bar-Natan algebra. We prove that $\unicode{x0414}^\infty(T)$ itself is a tangle invariant, settling a conjecture from their work. We deduce this from an invariance statement for a \emph{matrix multifactorization} $\mathcal M(T)$, generalizing a construction of Ballinger to the coefficient ring $\mathbb Z[G]$. This multifactorization is well defined up to a notion of 1-homotopy equivalence. The proof rests on explicit formulas for special deformation retracts of Koszul matrix factorizations, which realize delooping in this setting and may be of independent interest.
Comments30 pages, 4 figures
