We introduce a class of virtual knot invariants, called chord index invariants, and develop a general framework for lifting them to more refined, higher-order invariants. Applying this procedure to the affine index polynomial, we obtain the first-order refinement, termed the R invariant, and the second-order refinement, the S invariant. We collectively refer to these as chord degree invariants. Their fundamental properties are analyzed, and potential applications are explored, including estimates of the minimal crossing number and the Gordian distance.
Seungbhin Ha (Wed,) studied this question.
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