The $V_1$- and $V_2$-polynomials of a long virtual knot

Abstract

We introduce two polynomial invariants $V_1(K;t)$ and $V_2(K;t)$ of a long virtual knot $K$, which generalize the degree-two finite type invariants $v_{2,1}$ and $v_{2,2}$ of Goussarov, Polyak, and Viro. We establish their fundamental properties and show that any pair of Laurent polynomials can be realized as $(V_1(K;t),V_2(K;t))$ for some long virtual knot $K$. While these polynomials are not finite type invariants of any degree with respect to virtualizations, their first derivatives at $t=1$ define finite type invariants of degree three. As an application, we obtain an explicit Gauss diagram formula for the $\alpha_3$-invariant.