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

TL;DR

We construct two polynomial invariants $V_1(K;t)$ and $V_2(K;t)$ for long virtual knots that extend the degree-two GPV invariants $v_{2,1}$ and $v_{2,2}$. These polynomials are well-defined, invariant, and realize any pair of Laurent polynomials, while their first derivatives at $t=1$ yield degree-three finite-type invariants and an explicit Gauss diagram formula for the $\alpha_3$-invariant. Unlike virtualization-based finite-type theory, $V_1$ and $V_2$ are not finite-type of any degree under virtualization but are degree-two under crossing changes; their behavior under $\Delta$-moves ties into lower bounds on $\Delta$-distance. In the welded knot setting, $V_1'(K;1)$ is invariant and satisfies $V_1'(K;1)=\alpha_2(K)+\alpha_3(K)$, connecting to normalized Alexander invariants and enabling a Gauss diagram formula for $\alpha_3$.

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 $α_3$-invariant.

Figures (13)

• Figure 2.1: Three descriptions of a long virtual knot
• Figure 2.2: The curve $\alpha_i$ on $\Sigma$
• Figure 2.3: Inwardly and outwardly linked chords
• Figure 3.1: Two Reidemeister moves III
• Figure 3.2: Six descriptions on Gauss diagrams

Theorems & Definitions (32)

• Theorem 3.1
• proof
• Definition 3.2
• Remark 3.3
• Example 3.4
• Lemma 4.1
• proof
• Lemma 4.2
• proof
• Theorem 4.3