Hexagonal knot mosaics

A hexagonal knot mosaic is a way to draw a knot on a hexagonal board. A hexagonal r-mosaic is a hexagonal board of radius r that is centred around a given hexagon, and a knot is a closed loop of string in three dimensional space. The picture above shows a knot drawn on a hexagonal 5-mosaic.

It turns out that it is possible to represent any knot inside a hexagonal knot mosaic by using a kit of 27 types of rotatable hexagonal tile, as shown above. A connection point on a tile is a point at which a green curve meets the edge of the tile, and the tiles are required to be assembled in such a way that every connection point on a tile touches a connection point on an adjacent tile.

The recent paper Bounding Crossing Number in Rectangular and Hexagonal Knot Mosaics by Hugh Howards, Jiong Li, and Xiaotian Liu defines three types of hexagonal r-mosaics, which the authors call “standard”, “semi-enhanced”, and “enhanced”. The 3-mosaic shown above is classified as enhanced, because the yellow tile in its boundary (the outer ring of tiles) contains a crossing.

The other types of hexagonal r-mosaics are not allowed to contain crossings in their boundaries. A semi-enhanced r-mosaic, such as the one on the right of the above picture, is only allowed to use tiles 1, 2, 3, 4, 5, or 6 in its boundary. A standard r-mosaic, such as the one on the left, has the additional restriction that it is not allowed to use tile 6 in its boundary.

The crossing number of a knot is the number of times the string passes over or under itself. The knot mosaics below have been shaded so that the yellow, pink, and cyan tiles contain one, two, and three crossings respectively. This makes it easy to see that the knots on the left, middle, and right of the picture below have 19, 20, and 21 crossings, respectively. 

One of the main results of the paper by Howards, Li, and Liu is an answer to the question “What is the biggest crossing number that can be achieved by projecting a knot into a hexagonal r-mosaic?” For example, for enhanced r-mosaics with r ≥ 3, the authors prove that the biggest possible crossing number is 9r² − 25r + 15. This works out as 21 when r=3, which shows that the enhanced 3-mosaic in the picture has the maximal possible number of crossings. There are similar formulae for standard and semi-enhanced r-mosaics, all with a leading term of 9r². The standard 3-mosaic with 19 crossings and the semi-enhanced 3-mosaic with 20 crossings are also the best possible results in those settings.

Another result in the paper is a simpler proof of the corresponding theorem for rectangular mosaics on an r by r grid with r > 3. This result was first proved by Hugh Howards and Andrew Kobin in 2018, and it states that the maximum number of crossings is given by (r − 2)² − 2 if r is odd, and (r − 2)² − (r − 3) if r is even. The picture above shows a rectangular 6-mosaic that achieves the maximum number of crossings, 13, indicated by the yellow squares.

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Picture credits and relevant links

All the pictures are edited versions of pictures in the paper by Hugh Howards, Jiong Li, and Xiaotian Liu.

Here’s Wikipedia’s entry on knot theory.

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