AI 中文总结
本文证明了模4版本的川内猜想，该猜想基于有限型不变量的模式提出，借助Claude Fable 5，解决了双向纽结康威多项式分解在模4意义下的问题。
AI 中文摘要
川内猜想，对于双向纽结，其康威多项式可分解为$\nabla_K(z)=f(z)f(-z)$，其中$f(z)$为整系数多项式。川内与哈特利证明了强双向纽结的情况，哈特利利用纽结外部的JSJ分解将其推广到所有负双向纽结。厄莫蒂 - 洪格勒 - 韦伯首次给出一般情况的反例。2006年作者基于有限型不变量中的某些模式，猜想了一个与双向纽结的$\nabla_K \equiv f(z)f(-z) \pmod 4$等价的命题。本文借助Claude Fable 5证明了川内原始猜想的模4版本。
英文摘要
Kawauchi conjectured that the Conway polynomial of an amphicheiral knot factors as $\nabla_K(z)=f(z)f(-z)$ for some integer polynomial $f(z)$. In joint work with Hartley, he showed this was true for strongly amphicheiral knots, and Hartley used the JSJ decomposition of the knot exterior to generalize to all negative amphicheiral knots. Ermotti--Hongler--Weber were the first to publish a counterexample to the general case. Independently, in 2006 the author had conjectured a statement which is equivalent to the statement that $\nabla_K \equiv f(z)f(-z) \pmod 4$ for amphicheiral knots, based on certain patterns he noticed in finite type invariants. In this paper we prove this mod 4 version of Kawauchi's original conjecture as a consequence of a stronger integral statement. The mathematical content of this paper was produced with the help of Claude Fable 5.
CommentsUpdated main theorem to stronger integral criterion of positive amphicheiral knots
