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

TL;DR

This paper introduces two new polynomial invariants for long virtual knots, generalizing known finite type invariants, and explores their properties, realizability, and relation to finite type invariants of degree three.

Contribution

It defines the $V_1$ and $V_2$ polynomials for long virtual knots, establishes their properties, and links their derivatives to degree three finite type invariants.

Findings

Any pair of Laurent polynomials can be realized as $(V_1,V_2)$ for some knot.

The polynomials are not finite type invariants under virtualization.

Derivatives at $t=1$ yield degree three finite type invariants.

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.