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Theoretical analysis demonstrates a polynomial invariant for virtual knots via zero-smoothing, highlighting its capacity to distinguish complex knot diagrams.
In the present paper, a polynomial invariant [Formula: see text] for virtual knots is constructed by 0-smoothing at all real crossings and using the polynomial invariant [Formula: see text] for flat virtual knots. We further discuss some properties of the polynomial invariant [Formula: see text]. Finally, we construct a family of diagrams for virtual knots which can be distinguished from each other by invariant [Formula: see text].
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Liang et al. (2026) studied this question.
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