Birack modules and their link invariants (Q2853990)

Racks and quandles are algebraic structures whose axioms come from Reidemeister moves in Knot theory. They have been used extensively to construct invariants of knots and links. Rack modules were introduced by \textit{N. Andruskiewitsch} and \textit{M. Graña} [Adv. Math. 178, No. 2, 177--243 (2003; Zbl 1032.16028)] in the study of Nichols algebras.NEWLINENEWLINE The paper under review extends the notion of rack module to the case of biracks. This allows the authors to define birack module enhancements of the counting invariant generalizing the counting invariant defined in [\textit{A. Haas} et al., Osaka J. Math. 49, No. 2, 471--488 (2012; Zbl 1245.57009)]. The authors give examples showing that the new invariant is stronger than the unenhanced invariant. They show examples of virtual knots that are not distinguished by the Jones polynomial but are distinguished by the enhanced invariant. They also provide an example of a pair of knots that is distinguished by this invariant but not by the Alexander polynomial.