Link/knot invariants are series with integer coefficients, and it is a long-standing problem to get them positive and possessing cohomological interpretation. Constructing positive “superpolynomials” is not straightforward, especially for colored invariants. A simpler alternative is a multi-parametric generalization of the character expansion, which leads to colored “hyperpolynomials”. The third construction involves branes on resolved conifolds, which gives rise to still another family of invariants associated with composite representations. We revisit this triality issue in the simple case of the Hopf link and discover a previously overlooked way to produce positive colored superpolynomials from the DIM-governed four-point functions, thus paving a way to a new relation between super- and hyperpolynomials.

链接/结不变式是具有整数系数的级数，要使它们为正并具有同调解释是一个长期存在的问题。构造正的“超多项式”并非易事，特别是对于有色不变式。一个更简单的替代方法是字符扩展的多参数概括，从而产生彩色的“超多项式”。第三种构造涉及解析的凸线上的分形，这又产生了与复合表示相关的另一类不变式。我们在Hopf链接的简单情况下重新审视了这个问题，并发现了以前被忽略的从DIM控制的四点函数生成正色超多项式的方法，从而为超多项式和超多项式之间的新关系铺平了道路。