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In knot theory, a quandle cocycle invariant was defined. Moreover, to virtual knot theory it was generalized by avoiding for virtual crossings.

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Are there many application of a quandle cocycle invariant for virtual knots? For example, in surface knot theory, we can study an invertible, triple points number ets.

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To find the answer to your question, I would look through papers of Sam Nelson and his students, all of which are available on the arXiv. It is true that most of his work has to do with using quandles and biquandle colorings, he might have some work on cocycles.

Meanwhile, be aware that the cocycle invariants for classical knots can distinquish knots that have the same number of colorings. So there is a lot of potential for virtual knots as well. For example, in the virtual world you should be able to construct two different virtual knots that have the same number of colorings but which represent non-homologous 2-cycles. Also be aware of the database at Masahico Saito's homepage.

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