Cocycle invariants and oriented singular knots (Q821539)

In this paper the authors propose an enhancement of the singquandle counting invariant of singular knots and link analogous to the quandle 2-cocycle invariants for classical knots and links. As in the classical case, a weight function valued in an abelian group is defined at classical and singular crossings in a singquandle-colored diagram, with the condition for invariance obtained from analysis of the singquandle-colored Reidemeister moves. The authors refer to the condition as the ``2-cocycle condition'' by analogy with the classical case despite the lack of an apparent homology theory; it would be interesting to see further development of a homology theory fitting this scenario. Several examples are given to show the invariants of singular knots obtained from the construction.