New polynomial invariants differentiate virtual doodles, highlighting relationships among knot diagrams.
In this paper, we introduce new polynomial invariants of virtual doodles obtained by combining a state model with an integer labeling to flat virtual knot diagrams. Using our invariants we show that the doodle diagram d₄.₁ is different from the doodle diagram d₄.₄.
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Kim et al. (2026) studied this question.
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