Properties

Non Prime, Non Reduced or Non Alternating Diagrams

If the given diagram is not prime, reduced and alternating, we only provide the writhe of the diagram and the number of crossings.

Prime, Reduced and Alternating Diagrams

Class, Master Array, Zero Position Gauss Code

For a description of class, master array and zero position Gauss code, please refer to the preprint page. The master array is the complete invariant for alternating links used in enumerating the prime alternating links.

Archive, Alexander-Briggs and Thistlethwaite Numbers

These are described in detail here.

Orientations

The number of orientations of the knot/link up to mirror image. This number is computed in real time, if there are at least 1024 orientations, we terminate the algorithm so as not to overload the server.

Symmetry Group

The symmetries of an alternating, reduced, prime knot/link are visible up to flype in diagram. Clicking on the link will give the group as the set of permutations on the crossings of the knot/link. When given a prime, alternating, reduced knot, we identify the group as either cyclic or dihedral.

Unoriented Chirality, Invertibility

When given a prime, alternating, reduced knot, we give the following properties: (non-)invertible and one of chiral, +amphicheiral, -amphicheiral or fully amphicheiral. When given a prime, alternating, reduced link, we only give one of unoriented amphicheiral or chiral.

Oriented Chirality

We examine the orientations computed from the counting process of oriented amphicheirality. There are several possible outputs, if the given orientation is amphicheiral we return that immediately, otherwise we search for an amphicheiral orientation in the computed orientations and report back our findings. Of course, as stated above we terminate at 1024 orientations, so the search may not be exhaustive, but we at least report if we have found one in the first 1024 orientations.