关于辫子逆问题的一些注记
Some notes on the inverse problem for braids
AI总结：
本文研究两股辫子的逆问题，通过Kontsevich积分不变量建立与数论、数值分析和物理场论的联系，并为多股辫子情形提供视角。
摘要：
Kontsevich积分 $Z$ 为每个辫子 $b$（或更一般地，纽结 $k$）赋予位于有限维向量空间中的不变量 $Z_i(b)$，其中 $i = 0, 1, 2, ...$。除了特殊情况外，这些值尚不为人所知。逆问题是从不变量 $Z_i(b)$ 确定 $b$。本文研究两股辫子的情形，这已足以产生有趣且出乎意料的数学。特别地，我们发现了与数论、数值分析和物理场论的联系。然而，我们将以着眼于更一般的 $n$ 股辫子情形来进行这项研究。我们预期即使对于 $n=3$，解决逆问题也将面临真正的困难。本文中的大多数概念也适用于纽结，但为简化阐述，我们很少提及。本文的组织和大部分写作早于其最重要的结果。我们希望稍后能更好地展示并进一步发展这些结果。
英文摘要：
The Kontsevich integral $Z$ associates to each braid $b$ (or more generally knot $k$) invariants $Z_i(b)$ lying in finite dimensional vector spaces, for $i = 0, 1, 2, ...$. These values are not yet known, except in special cases. The inverse problem is that of determining $b$ from its invariants $Z_i(b)$. In this paper we study the case of braids on two strands, which is already sufficient to produce interesting and unexpected mathematics. In particular, we find connections with number theory, numerical analysis and field theory in physics. However, we will carry this study out with an eye to the more general case of braids on $n$ strands. We expect that solving the inverse problem even for $n=3$ will present real difficulties. Most of the concepts in this paper also apply to knots, but to simplify the exposition we will rarely mention this. The organisation and bulk of the writing of this paper predates its most significant results. We hope later to present better and develop further these results.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
