Minimal grid diagrams of the prime knots with crossing number 14 and arc index 13

Abstract: There are 46,972 prime knots with crossing number 14. Among them 19,536 are alternating and have arc index 16. Among the non-alternating knots, 17, 477, and 3,180 have arc index 10, 11, and 12, respectively. The remaining 23,762 have arc index 13 or 14. There are none with arc index smaller than 10 or larger than 14. We used the Dowker-Thistlethwaite code of the 23,762 knots provided by the program Knotscape to locate non-alternating edges in their diagrams. Our method requires at least six non-alternating edges to find arc presentations with 13 arcs. We obtained 8,027 knots having arc index 13. We show them by their minimal grid diagrams. The remaining 15,735 prime non-alternating 14 crossing knots have arc index 14 as determined by the lower bound obtained from the Kauffman polynomial.