We equip a knot $K$ with a set of colored bonds, that is, colored intervals properly embedded into $\mathbb{R}^3 \setminus K$. Such a construction can be viewed as a structure that topologically models a closed protein chain including any type of bridges connecting the backbone residues. We introduce an invariant of such colored bonded knots that respects the HOMFLYPT relation, namely the HOMFLYPT skein module of colored bonded knots. We show that the rigid version of the module is freely generated by colored $\Theta$-curves and handcuff links, while the non-rigid version is freely generated by the trivially embedded $\Theta$-curve. The latter module, however, does not provide information about the knottedness of the bonds.

中文翻译：

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彩色粘合结的不变量

我们为一个结 $K$ 配备一组彩色键，即正确嵌入 $\mathbb{R}^3 \setminus K$ 的彩色区间。这种结构可以被视为一种结构，该结构对封闭的蛋白质链进行拓扑建模，包括连接主链残基的任何类型的桥。我们引入了一种尊重 HOMFLYPT 关系的彩色粘合结的不变量，即彩色粘合结的 HOMFLYPT 绞线模块。我们表明模块的刚性版本是由彩色 $\Theta$-curves 和手铐链接自由生成的，而非刚性版本是由普通嵌入的 $\Theta$-curve 自由生成的。然而，后一个模块不提供有关债券打结的信息。