# Grid Diagrams

Every knot can be drawn on a grid, as illustrated above for the knot 6_2. Such a grid diagram drawn on an N by N grid can be described by a set of 2N grid points (a,b) where a and b are integers between 1 and N. The points must satisfy the conditions that each row and each column in the grid contains exactly two points. The knot is formed by joining the grid points with horizontal and vertical segments, with the vertical segments always passing over the horizontal segments. For the knot 6_2, the grid notation is formally:
[[1,1],[1,3],[2,2],[2,4],[3,3],[3,5],[4,4],[4,7],[5,6],[5,8],[6,1],[6,7],[7,5],[7,8],[8,2],[8,6]]

The minimal N for which a given knot K has a grid diagram of size N is called the grid number. The grid number of a knot is the minimal size of a grid diagram. It is also called the Arc Index.

Our data for grid numbers through 13 crossing was provided by Gyo Taek Jin, based in part on work in the such papers as [1, 2, 3, 4].

## References

[1] Gyo Taek Jin, Wang Keun Park, A tabulation of prime knots up to arc index 11, 2010 Arxiv reprint.

[2] Gyo Taek Jin, Hyuntae Kim, Seungwoo Lee, Hun Joo Myung, Prime knots with arc index 12 up to 16 crossings, 2020 Arxiv reprint.

[3] Gyo Taek Jin, Hwa Jeong Lee, Minimal Grid Diagrams of the Prime Alternating Knots with 12 Crossings, 2020 Arxiv reprint.

[4] Hwa Jeong Lee, Yoonsang Lee, Chanmin Lee, Yeseo Park, Hun Kim, Gyo Taek Jin, Minimal grid diagrams of the prime knots with crossing number 13 and arc index 13, 2024 Arxiv reprint.

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