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 15 or larger. We used the 24 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 nd arc presentations with 13 arcs. We obtained 7,504 knots having arc index 13. We show them by their minimal grid diagrams. It is a joint work with Hwa Jeong Lee, Alexander Stoimenow, Minchae Kim, Songwon Ryu, Dongju Shin, and Hun Kim.
Grid diagram
We can define something called a grid diagram. Here a knot diagram is drawn in horizontal lines. This can be used to define an arc presentation.
Out of a grid diagram we create a matrix around it. This is a Cromwell Matrix, i.e.it consists of 0s and 1s.
Molton-Bertrame, Park and the authors found a few connections between the crossing index and the arc index.
Construction of Cromwell matrices
The author describes how he calculated all of the arc indices along with their Cromwell matrices.
To create arc diagrams two procedures can be used. Either a Knot/spoke method or a filtered spanning tree method.