The Knot Table

By clicking at this site you may see all distinct knots with up to 12 crossings. This table has been made through a direct conversion of the computer output. We now explain the meaning of the various symbols.

We start first with the list of knots. For each knot, the first line denotes its referrence number, starting from 0 for the trivial knot, and ending to 2977 for the last 12 crossing knot. The second line shows the actual notation of the knot, and is written in italics. The meaning of the notation has been explained in a previous page. The following lines denote the HOMFLYPT polynomials as follows: a term c(m,n) indicates the existence of a term c times x**m times y**n in the HOMFLYPT polynomial.

Once all knots have thus been presented, we continue by showing how knots with identical polynomials are distinguished. First, we list the groups of knots sharing the same (or symmetrical) polynomials. We continue with the colorization invariants. We only list the invariants and the groups of knots where distinctness is shown through unequal number of coloring possibilities. Each colorization is presented through a partition of some positive integer, indicating that the colorization used corresponds to a conjugacy class of the permutation group that corresponds to such a partition. Knots here are presented through their referrence numbers, which were given before.

One may check that at the end of the file, all knots have been shown distinct to each other either through the HOMFLYPT polynomials, or through some colorization invariant.