Find-A-Knot (beta)

Enter knot data

Notation: PD DT braid

Example Inputs:

Example of PD notation for a connected sum

input: [[5,9,6,8],[7,13,8,12],[9,15,10,14],[11,7,12,6],[13,11,14,10],[15,21,16,20],[17,2,18,3],[19,17,20,16],[21,5,0,4],[1,18,2,19],[3,1,4,0]]

Example of DT notation for a prime knot

input: [4, 8, 10, 2, 6]

Example of braid notation for a prime knot

input: [1,1,1,2,-1,2]

Output format:

For each prime summand of the found knot, we give a 4-tuple of the following entries.

The name of the knot
the symmetry version of the knot
the symmetry type of the knot (see below for more information)
The index of the knot in the full list of 16 crossing prime knots (available as a KnotInfo download).

For instance, ('K3a1', 'mirror', 'reversible', 1) is the mirror image of the trefoil knot K3a1, which is a reversible knot and is the first knot in the list.

Symmetry Type:

Given an oriented diagram for a knot K, one can associate four knots. The knot itself, "identity" K; the "mirror" m(K) formed by changing all the crossings; the "reverse" r(K) formed by changing the direction of the knot; and the "mirror reverse" m(r(K)). There are five symmetry types.

If K, m(K), r(K), and m(r(K)) are all distinct, then K is chiral.
If all are the same, then K is fully amphicheiral.
If K = r(K) and no other, then K is reversible.
If K = m(r(K)) and no other, then K is negative amphicheiral.
If K = m(K) and no other, then K is positive amphicheiral.

Authors:

This is a (draft) implementation of a Python program "Find-A-Knot" being developed by Chuck Livingston and Ana Wright.