An Index Refined Winding Pair Polynomial for Planar Knotoids

Planar knotoids contain endpoint position information because the forbidden endpoint moves prevent the leg and the head from passing across arcs. Existing polynomial invariants based on Gauss diagrams and winding data record important parts of this information, but they may separate endpoint winding data from affine index data or combine winding contributions only after summation. We introduce an index refined winding pair polynomial in three variables for oriented planar knotoids. At each crossing, the invariant records the ordered winding pair of the crossing lobe together with the affine index weight of the same crossing. We prove invariance under planar knotoid equivalence, derive formulas for orientation reversal, mirror image and planar product, and show that the invariant is a Vassiliev invariant of degree one. We also obtain lower bounds for crossing number, Gordian distance and unknotting number from a nonconstant coefficient norm. Finally, explicit computations show that the winding signed sum polynomial and the affine index polynomial, even when considered together, do not determine the new invariant.