An invariant for colored bonded knots

Abstract

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.