A New Polynomial for Checkerboard-Colorable 4-Valent Virtual Graphs

Abstract

We assign a new polynomial to any checkerboard-colorable 4-valent virtual graph in terms of its Euler circuit expansion. This provides a new combinatorial formulation of the Kauffman-Jones polynomial for checkerboard-colorable virtual links.

Figures (15)

• Figure 1: Checkerboard-colored 4-valent virtual graph.
• Figure 2: Allowed orientations around each vertex for a 2-digraph to have a source target structure.
• Figure 3: Determining a checkerboard coloring of a 4-valent graph from a source target structure and vice versa.
• Figure 4: A 2-digraph with an Euler circuit $\gamma = v_{1}e_{1}v_{2}e_{2}v_{4}e_{3}v_{1}e_{4}v_{2}e_{5}v_{3}e_{6}v_{4}e_{7}v_{3}e_{8}v_{1}$ on the left and the corresponding chord diagram of $\gamma$ on the right.
• Figure 5: A checkerboard-colorable virtual link diagram.

Theorems & Definitions (8)

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• proof : Proof of Theorem \ref{['theorem1']}
• proof : Proof of Theorem \ref{['theorem2']}