On a Generalization of Alexander Polynomial for Long Virtual Knots

We construct new invariant polynomial for long virtual knots. It is a generalization of Alexander polynomial. We designate it by ζ meaning an analogy with ζ-polynomial for virtual links. A degree of ζ-polynomial estimates a virtual crossing number. We describe some application of ζpolynomial for the study of minimal long virtual diagrams with respect number of virtual crossings. Virtual knot theory was invented by Kauffman around 1996 [Ka1]. Long virtual knot theory was invented in [GPV] by M. Goussarov, M. Polyak, and O. Viro. ζ-polynomial for virtual link was introduced independently by several authors (see [KR],[Saw],[SW],[Ma1]), for the proof of their coincidence, see [BF]. The idea of two types of classical crossings in a long diagram, which were called ◦ (circle) and ∗ (star), was invented by V.O. Manturov (see [Ma4],[Ma3]). In present paper we called ◦ and ∗ crossings by early overcrossing and early undercrossing respectively. To consider early overcrossings and early undercrossings is the basis idea for a construction of ζ-polynomial in the case of long virtual knots. Definition 1.1. By a long virtual knot diagram we mean a smooth immersion f : R → R such that: 1) outside some big circle, we have f(t) = (t, 0); 2) each intersection point is double and transverse; 3) each intersection point is endowed with classical (with a choice for underpass and overpass specified) or virtual crossing structure. Definition 1.2. A long virtual knot is an equivalence class of long virtual knot diagrams modulo generalized Reidemeister moves. Definition 1.3. By an arc of a long virtual knot diagram we mean a connected component of the set, obtained from the diagram by deleting all virtual crossings (at classical crossing the undercrossing pair of edges of the diagram is thought to be disjoint as it is usually illustrated).