A quick note on homsets and diagrams

The homset invariant of a knot or link [Formula: see text] with respect to an algebraic knot coloring structure [Formula: see text] can be identified with a set of colorings of a diagram of [Formula: see text] by elements of [Formula: see text] via an identification of diagrammatic generators with algebraic generators. In some cases, particularly when either [Formula: see text] or [Formula: see text] has a high degree of symmetry, distinct homset elements can be represented by superficially similar [Formula: see text]-colored diagrams, potentially leading to confusion. In this paper, we examine and attempt to clarify this phenomenon.