链环的卫星与不变量
Satellites and invariants of links
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中文总结 AI 辅助
本文肯定地解决了Arnold-Moffatt纲领中关于存在非环绕数函数的可缆化有限型链环不变量的问题，并推广至任意卫星，通过研究Conway位势函数的低次系数完成证明。
中文摘要 AI 辅助
一个关于$m$分量链环的不变量$v$被称为“可缆化的”，如果存在一个$k$，使得每当从链环$L=(K_1,\dots,K_m)$通过将每个纽结$K_i$替换为其$(p_i,q_i)$-缆绳（对于某些$p_i$和$q_i$）而得到链环$L'$时，我们有$v(L')=(p_1\cdots p_m)^kv(L)$。以下问题隐含在P. M. Akhmetiev的多篇论文中，并源自Arnold-Moffatt纲领，该纲领旨在寻找磁场能量的拓扑下界：在$S^3$中，是否存在一个可缆化的有限型链环不变量，它不是两两环绕数的函数？我们肯定地解决了这个问题。此外，我们表明缆绳可以被任意卫星所替代。证明的很大一部分是对Conway位势函数$\Omega_L(x_1,\dots,x_n)$展开为Conway变量$z_i=x_i-x_i^{-1}$的形式幂级数时的低次系数进行研究。我们还讨论了“模$n-1$型不变量可缆化”的$n$型不变量，一些不是有限型的可缆化不变量（特别是Milnor的$\bar\mu$-不变量的一种特定修改），以及它们在螺线管链环上的应用。
英文摘要
An invariant $v$ of $m$-component links is called "cableable" if there exists a $k$ such that whenever a link $L'$ is obtained from a link $L=(K_1,\dots,K_m)$ by replacing each knot $K_i$ with its $(p_i,q_i)$-cable for some $p_i$ and $q_i$, we have $v(L')=(p_1\cdots p_m)^kv(L)$. The following problem is implicit in a number of papers by P. M. Akhmetiev and originates from the Arnold-Moffatt program for finding topological lower bounds for the energy of a magnetic field: Does there exist a cableable finite type invariant of links in $S^3$ which is not a function of the pairwise linking numbers? A potential solution of this problem was proposed by Akhmetiev himself, with the desired invariant defined as an analytic expression involving a magnetic field modeled on the given link, but we note that basic properties that he claimed of his invariant cannot be all true. In any case, we offer a different solution, with the desired invariant being a function of the coefficients of the Conway polynomial of the link and its sublinks. Moreover, we show that the cables can be replaced by arbitrary satellites. Much of the proof is a study of low degree coefficients of the Conway potential function $Ω_L(x_1,\dots,x_n)$ expanded as a formal power series in Conway's variables $z_i=x_i-x_i^{-1}$. We also discuss type $n$ invariants which are "cableable up to an invariant of type $n-1$", some cableable invariants which are not of finite type (particularly a certain modification of Milnor's $\barμ$-invariants), and applications to links of solenoids.
发表机构
- Steklov Mathematical Institute of Russian Academy of Sciences（俄罗斯科学院斯捷克洛夫数学研究所）
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