AI 中文总结
研究 I 型李超代数纽结多项式，证明其\(t\)次至多为奇根数乘纽结亏格，通过\(q = 1\)特化得到互补界，在亚历山大多项式能检测亏格时两界成等式。
AI 中文摘要
由 I 型李超代数（除\(\mathfrak{psl}(n|n)\)外）的典型表示所着色的纽结多项式有两个变量\(q\)和\(t\)，后者对应于特殊奇根的复值权重。我们证明，对于 I 型李超代数的每个典型表示，纽结多项式的\(t\)次至多为奇根数量乘以纽结的亏格。通过在\(q = 1\)处的特化可得到一个互补界，即至少为奇根数量乘以亚历山大多项式的次数。当亚历山大多项式能检测纽结的亏格时，如交错纽结和纤维化纽结的情况，这两个界成为等式。
英文摘要
Knot polynomials colored by typical representations of Lie superalgebras of type I (except $\mathfrak{psl}(n|n)$) have two variables $q$ and $t$, the latter corresponding to the complex-valued weight of the distinguished odd root. We prove that for every typical representation of a Lie superalgebra of type I, the $t$-degree of the knot polynomial is at most the number of odd roots times the genus of the knot. A complimentary bound being at least the number of odd roots times degree of the Alexander polynomial can be obtained from a specialization at $q=1$. These two bounds become equalities when the Alexander polynomial detects the genus of the knot, as is the case for alternating knots and fibered knots.
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