论文标题
亚历山大多项式的缺陷和程度
Defect and degree of the Alexander polynomial
论文作者
论文摘要
缺陷表征了结多项式的差分（环形）扩展中项的分解深度，即Chern-Simons理论中的非扰动Wilson平均值。我们证明，可以将缺陷替代地描述为基本亚历山大多项式的$ q^{\ pm 2} $中的程度，这正式与无颜色的情况相对应。我们还提出了一个问题，如果这些亚历山大多项式可以是给定程度的任意整数多项式。回答后一个问题的首次尝试是对2链圆环结的反平行后代的初步分析，这为所有缺陷的所有值提供了一组不错的示例。在缺陷零结的情况下，答案是积极的，在扭结的情况下可以观察到的。这一证明的猜想还允许我们为对称颜色的亚历山大多项式用于缺陷零提供一组$ c $ - 多项式。在这种情况下，我们实现了表示和结变量的完整分离。
Defect characterizes the depth of factorization of terms in differential (cyclotomic) expansions of knot polynomials, i.e. of the non-perturbative Wilson averages in the Chern-Simons theory. We prove the conjecture that the defect can be alternatively described as the degree in $q^{\pm 2}$ of the fundamental Alexander polynomial, which formally corresponds to the case of no colors. We also pose a question if these Alexander polynomials can be arbitrary integer polynomials of a given degree. A first attempt to answer the latter question is a preliminary analysis of antiparallel descendants of the 2-strand torus knots, which provide a nice set of examples for all values of the defect. The answer turns out to be positive in the case of defect zero knots, what can be observed already in the case of twist knots. This proved conjecture also allows us to provide a complete set of $C$-polynomials for the symmetrically colored Alexander polynomials for defect zero. In this case, we achieve a complete separation of representation and knot variables.
