Classification of knots and links with mosaic number 6

Abstract

Mosaic number is one of the knot invariants. However, there are not so many results even in the case of the prime knots. Recent studies completed the classification of prime knots with mosaic number 7 or less, but those are almost all the consequences until now. In this paper, we classify prime knots with mosaic number 6 in slightly different method. Compared to the preceding researches, the approach used in our work is a bit cumbersome in some point, but it also has some advantages. Such differences and advantages will be pointed out during the classification. Also, we give some partial results and conjectures on the classifications of composite knots and links.

Consequently, there are 122 prime knots with mosaic number 6 and crossing number greater than 8. Moreover, we may discover a mild clue that there are not so many composite knots and links with mosaic number 6. The main difficulty is that there are enormous amount of possible knot projections on the 6 x 6 knot mosaic, even though we exclude considerable configurations in advance. It was handled by a proper software package with some additional computer programming.