Find-A-Knot (beta)

Enter knot data

Notation: PD DT braid

Sample input:

PD (connected sum): [[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]]

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

braid: [1,1,1,2,-1,2]

Output format: For each prime summand of the found knot, we give a 4-tuple: The name of the knot; the symmetry version of the knot; the symmetry type of the knot; and its index 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.

Comments

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

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.