Tangle embeddings

The number of Fox colorings, as well as branched coverings and quantum invariants, were used as obstructions to tangle embeddings (see, for example, [Kre99,KSW00,PSW04*,Rub00]). Quandle cocycle invariants can be used as obstructions as well [AERSS].

For a tangle $T$, the cocycle invariant $\Phi_\phi(T)$ is defined by the formula similar to the case of knots, where, for the purpose of applications to tangle embeddings, we require that the colors on the boundary points of $T$ are monochromatic (have the same color). Such a coloring is called boundary monochromatic.

Suppose a tangle diagram $T$ embeds in a knot (or link) diagram $D$. For any boundary monochromatic coloring ${\mathcal C}$ of $T$, there is a unique coloring $\tilde{\mathcal C}$ of $D$ such that the restriction on $T$ is the given coloring of $T$ ( $\tilde{\mathcal C}\vert _T={\mathcal C}$) and that all the other arcs (the arcs of $D$ outside of $T$) receive the same color as that of the boundary color of $T$. Then the contribution of $\tilde{\mathcal C}$ to $\Phi_\phi(K)$ agrees with the contribution of ${\mathcal C}$ to $\Phi_\phi(T)$, since the contribution from the outside of the tangle is trivial, being monochromatic. Hence we obtain the condition $\Phi_\phi(T) \subset_m \Phi_\phi(K)$, where the multi-subset $M \subset_m N$ is defined as follows. If an element $x$ is repeated $n$ times in a multiset, call $n$ the multiplicity of $x$, then $M \subset_m N$ for multisets $M$, $N$ means that if $x \in M$, then $x \in N$ and the multiplicity of $x$ in $M$ is less than or equal to the multiplicity of $x$ in $N$.

In [AERSS], tables of tangles [KSS03] and knots [Liv] were examined. For those in the tables, the cocycle invariants were computed, and compared to obtain information on which tangles do not embed in which knots in the tables. For one case it was possible to completely determine, using the cocycle invariants, in which knots up to $9$ crossings a tangle embeds.