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

TL;DR

This paper catalogs minimal grid diagrams for prime knots with crossing number 14, focusing on those with arc index 13 and 14, providing a comprehensive resource for knot classification.

Contribution

It presents the first extensive enumeration of minimal grid diagrams for prime knots with crossing number 14 and arc index 13 or 14.

Findings

Identified 8,027 knots with arc index 13

Identified 15,735 knots with arc index 14

Provided minimal grid diagrams for these knots

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 obtained 8,027 knots having arc index 13 and 15,735 knots having arc index 14. We show them by their minimal grid diagrams.