The study of knot mosaics is based upon representing knot diagrams using a set of
tiles on a square grid. This branch of knot theory has many unanswered
questions, especially regarding the efficiency with which we draw knots as
mosaics. While any knot or link can be displayed as a mosaic, for most of them
it is still unknown what size of mosaic (mosaic number) is necessary and
how many nonblank tiles (tile number) are necessary to depict a given knot
or link. We implement an algorithmic programming approach to find the
mosaic number and tile number of all prime knots with crossing number 10 or
less. We also introduce an online repository which includes a table of knot
mosaics and a tool that allows users to create and identify their own knot
mosaics.
Keywords
knot theory, knot mosaic, tile number, space efficient,
mosaic number, table of knots