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Cocycle invariants and oriented singular knots. (English) Zbl 1485.57013

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.

MSC:

57K12 Generalized knots (virtual knots, welded knots, quandles, etc.)
05C38 Paths and cycles
05A15 Exact enumeration problems, generating functions
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