Virtual shadow modules and their link invariants (Q2909614)

Virtual knot theory was introduced by \textit{L. Kauffman} in [Eur. J. Comb. 20, No. 7, 663--690 (1999; Zbl 0938.57006)] as a combinatorial generalization of classical knot theory. Roughly speaking, a virtual link diagram is a link diagram on the \(2\)-sphere with two types of crossings: real crossings, which are the same as crossings for classical link diagrams, and virtual crossings, which are of another type. A virtual knot is an equivalence class of virtual knot diagrams under the generalized Reidemeister moves. Kauffman was led to this theory by considerations including the problem of Gauss code reconstruction of knots and the study of knots in thickened surfaces.NEWLINENEWLINE In [\textit{J. S. Carter} et al., Osaka J. Math. 42, No. 3, 499--541 (2005; Zbl 1089.57008)], knot invariants were defined using the quandle module algebra proposed by \textit{N. Andruskiewitsch} and \textit{M. Graña} [Adv. Math. 178, No. 2, 177--243 (2003; Zbl 1032.16028)]. In the paper under review, the authors generalize the quandle module algebra by introducing and studying the virtual birack shadow algebra (an algebra satisfying certain diagrammatically motivated properties) and a twisted version of it. As an application, the authors use modules over these algebras to define (enhanced) invariants for virtual links. They show that these enhanced invariants can detect orientation reversal and are not determined by the knot group, the arrow or Miyazawa polynomials, or the twisted Jones polynomials. The paper ends with some open questions.