Tangle embeddings

The number of Fox colorings, as well as branched coverings and quantum invariants, was used as obstructions to tangle embeddings (see, for example, [PSW04*]). Cocycle invariants can be used as obstructions as well. We illustrate the method by an example. Consider the weight sum (the sum of cocycle values) of a tangle $T$ for a coloring such that the boundary points of the tangle are required to have the same color. If the tangle $T$ is a subtangle of a link $L$, such a coloring extends to a coloring of $L$ by choosing the color of the boundary points of $T$ for all arcs of $L$ outside of $T$. Then the weight sum for $L$ is the same as that of $T$ for these colorings. Hence the cocycle invariant of $L$ must have the same contribution.

Figure: A colored tangle

For example, see the colored tangle in Fig. , which has the weight sum $2 \in \mathbb{Z}_3$. Hence this does not embed in any link that does not have $2$ in its cocycle invariant with $R_3$.