Abstract. Given a virtual knot K, we introduce a new group-valued invariant VGK 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 0-th ideal is a polynomial HK (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 GK (s, t) introduced in [40,24,42]. 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 : VGK → GLn(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.

Abstract. Using Gauss diagrams, one can define the virtual bridge number vb(K) and the welded bridge number wb(K), invariants of virtual and welded knots satisfying wb(K) ≤ vb(K). If K is a classical knot, Chernov and Manturov showed that vb(K) = br(K), the bridge number as a classical knot, and we ask whether the same thing is true for welded knots. The welded bridge number is bounded below by the meridional rank of the knot group G K , and we use this to relate this question to a conjecture of Cappell and Shaneson. We show how to use other virtual and welded invariants to further investigate bridge numbers. Among them are Manturov's parity and the reduced virtual knot group G K , and we apply these methods to address Questions 6.1, 6.2, 6.3 & 6.5 raised by Hirasawa, Kamada and Kamada in their paper Bridge presentation of virtual knots, J. Knot Theory Ramifications 20 (2011), no. 6, 881-893. Virtual and welded knotsThe classical theory of knots involves studying embeddings of S 1 → S 3 up to ambient isotopy. Under projection to a generic two-plane, any given knot is mapped to an immersion S 1 R 2 with finitely many immersion points, each of which is a transverse double point. The 3-dimensional knot can be reconstructed from its projection by keeping track of crossing information at each double point. The knot projection, together with the crossing information, is called the knot diagram. Two knot diagrams correspond to isotopic knots if and only if they are related by a sequence of Reidemeister moves.To every oriented knot diagram corresponds a Gauss word, which records the order in which the crossings are encountered starting from a fixed basepoint on the knot, along with the sign of each crossing. The Gauss word is conveniently represented by a decorated trivalent graph called the Gauss diagram, consisting of a circle oriented counterclockwise together with n signed, directed arcs connecting n distinct pairs of points on the circle. The directed arcs, or chords, point from an overcrossing (arrowtail) to an undercrossing (arrowhead). The sign of the chord indicates whether the crossing is right-handed (positive) or left-handed (negative). The Reidemeister moves on knot diagrams can be translated into corresponding moves on the Gauss 2010 Mathematics Subject Classification. Primary: 57M25, Secondary: 57M27.