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Theoretical study reveals coloring-allowed invariants and four-phases functions for planar knotoids, highlighting new algebraic tools to classify open-ended knot curves.
In recent years, numerous polynomial invariants of knotoids have been constructed, some of which are defined with the signs of the crossings. In this paper, the coloring-allowed invariants of planar knotoids, which is a class of planar knotoid invariants defined with the coloring number are introduced. We demonstrate several basic properties of the coloring number and some examples of coloring-allowed invariants, among which the 4-phases functions are discussed in detail, including their invariance and properties.
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Chen et al. (2026) studied this question.
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