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 , the cocycle invariant
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
are monochromatic (have the same color).
Such a coloring is called boundary monochromatic.
Suppose a tangle diagram embeds in a knot (or link) diagram
.
For any boundary monochromatic coloring
of
,
there is a unique coloring
of
such that
the restriction on
is the given coloring of
(
) and that all the other arcs
(the arcs of
outside of
) receive the same color as that of the
boundary color of
.
Then the contribution of
to
agrees with the contribution of
to
,
since the contribution from the outside of the tangle is trivial, being
monochromatic.
Hence we obtain the condition
,
where the multi-subset
is defined as follows.
If an element
is repeated
times in a multiset, call
the
multiplicity of
, then
for multisets
,
means
that if
, then
and the multiplicity of
in
is less than or equal to the multiplicity of
in
.
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 crossings a tangle embeds.