A birack, a set with two binary operations which satisfy some constraints derived from the Reidemeister moves, is widely used in constructing invariants in knot theory. A basic knot invariant is the cardinality of the set of homomorphisms from the fundamental birack of a knot \(K\) to a given finite birack. In [R. Bauernschmidt and S. Nelson, Commun. Contemp. Math. 15, No. 3, Article ID 1350006, 13 p. (2013; Zbl 1290.57018)], a sort of generalization of the counting invariant was defined. The paper under review offers another enhancement of the counting invariant via an algebraic structure called birack dynamical cocycle.
More precisely, for a fixed birack coloring the authors introduce a set \(S\) with maps \(D_{x, y}:S\times S\rightarrow S\times S\), here \(x, y\) are two elements of the birack. Then by assigning elements of \(S\) to the semiarcs of the colored knot diagram one can associate a weight, the number of assignments to the fixed birack coloring. In this way the authors define the birack dynamical cocycle enhanced polynomial, which is an enhancement of the counting invariant. The authors give some examples showing that the new invariant is stronger than the unenhanced counting invariant.