Pseudoknots are generalizations of knots where, in addition to classical over/under crossings, the diagrams of pseudoknots also present crossings of undetermined nature, i.e. which are not known to be over or under crossings. Such a situation can be imagined in a probabilistic manner, where we associate to such an undetermined crossing a probabiliy of it being an over-crossing, or an under-crossing. The study of such structures arises rather naturally in certain biological situations, such as the study of DNA, where experimental uncertainty results in undetermined combinatorial information. The isotopy classes of pseudoknots are described by a set of Reidemeister moves where precrossings appear as well. These moves are the same as those for singular knots, except for a single move.
The article under review utilizes a ternary operation called psybracket, similar to the notion of Niebrzydowski tribracket, with some extra axioms to introduce certain coloring (counting) invariants of pseudoknots. The authors show that the defining axioms correspond to the combinatorial moves that characterize isotopy classes of pseudoknots upon defining a notion of region coloring of a diagram of a pseudoknot. The article presents extensive examples and computations to showcase the theory introduced. Finally, the authors provide a compendium of questions for future perspectives on the subject.