Alexander invariants for virtual knots

Given a virtual knot K, we introduce a new group-valued invariant VG K called the virtual knot group, and we use the elementary ideals of VG K to define invariants of K called the virtual Alexander invariants. For instance, associated to the zeroth ideal is a polynomial H K (s, t, q) in three variables which we call the virtual Alexander polynomial, and we show that it is closely related to the generalized Alexander polynomial G K (s, t) introduced by Sawollek; Kauffman and Radford; and Silver and Williams. We define a natural normalization of the virtual Alexander polynomial and show it satisfies a skein formula. We also introduce the twisted virtual Alexander polynomial associated to a virtual knot K and a representation ϱ : VG K → GL n (R), and we define a normalization of the twisted virtual Alexander polynomial. As applications we derive bounds on the virtual crossing numbers of virtual knots from the virtual Alexander polynomial and twisted virtual Alexander polynomial.