AI 中文总结
研究S. Lando提出的\(\mathfrak{sl}(2)\)权重系统能否扩展为图的4-不变量这一问题，通过构建递推关系等方法，发现一般答案为否，但某些特殊化存在扩展，还研究了多项式系数及扩展问题。
AI 中文摘要
S. Lando长期以来的一个问题是，\(\mathfrak{sl}(2)\)权重系统是否能扩展为图的唯一4-不变量。我们表明，一般情况下，这个问题的答案是否定的。不过，对于权重系统的某些特殊化，扩展是存在的。计算权重系统的两种此类特殊化的显式公式已知。我们为另一种这样的扩展构建了递推关系，并讨论了最后剩下的特殊化，推测它允许扩展。我们还研究了\(\mathfrak{sl}(2)\)权重系统的多项式系数并解决了其扩展问题。
英文摘要
A long-standing question by S. Lando asks whether the $\mathfrak{sl}(2)$-weight system extends to a unique 4-invariant of graphs. We show that, in full generality, the answer to this question is negative. However, for certain specializations of the weight system, extensions do exist. Explicit formulae for computing two such specializations of the weight system are already known. We construct recurrence relations for one additional such extension and discuss the last remaining specialization, which conjecturally admits an extension. We also study the polynomial coefficients of the $\mathfrak{sl}(2)$-weight system and resolve the questions concerning their extension.
Comments14 pages, 5 figures
