On an invariant for colored classical and singular links

A colored link, as defined by Francesca Aicardi, is an oriented classical link together with a ‘coloration’, which is a function defined on the set of link components and whose image is a finite set of ‘colors’. An oriented classical link can be regarded as a colored link with its components colored with a sole color. Aicardi constructed an invariant F ( L ) F(L) of colored links L L defined via skein relations. When the components of a colored link are colored with the same color or when the colored link is a knot, F ( L ) F(L) is a specialization of the HOMFLY-PT polynomial. Aicardi also showed that F ( L ) F(L) is a stronger invariant than the HOMFLY-PT polynomial when evaluated on colored links whose components have different colors. In this paper, we provide a state-sum model for the invariant F ( L ) F(L) of colored links using a graphical calculus for oriented, colored, 4-valent planar graphs. We also extend F ( L ) F(L) to an invariant of oriented colored singular links.