Tangle embeddings and quandle cocycle invariants. (English) Zbl 1180.57006

The authors study the embeddings of tangles in knots. A method using quandle cocycle invariants as obstructions to embeddings of oriented tangles in oriented knots is given. For a given tangle diagram or link diagram, and a quandle 2- or 3- cocycle with values in an Alexander quandle, the quandle cocycle invariant is the multiset, indexed by the set of all colorings, of the sums of the Boltzmann weights of a coloring at all crossings of the tangle.
It is shown that if a tangle embeds in a link, then the quandle cocycle invariant of the tangle is included in the quandle cocycle invariant of the link. Computations are carried out for a table of some oriented prime tangles of 6 and 7 crossings and knots up to 9 crossings to investigate which tangles may or may not embed in which knots. The general scheme of the computation is illustrated in two examples using quandle cocycle invariant of tangles and knots and the above-mentioned theorem. Quandle cocycle invariants for disjoint tangles are similarly defined, and a similar result for a disjoint union of tangles which embeds in a link is proved.


57M25 Knots and links in the \(3\)-sphere (MSC2010)


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