Alexander invariants for virtual knots

作者：Boden Hans U^*; Dies Emily; Gaudreau Anne Isabel; Gerlings Adam; Harper Eric; Nicas Andrew J

来源：Journal of Knot Theory and Its Ramifications, 2015, 24(3): 1550009.

DOI：10.1142/S0218216515500091

• Virtual knots
• knot groups
• Alexander invariants
• twisted Alexander polynomials
• virtual crossing number

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 VGK 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 rho : 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.

• 出版日期2015-3