Find-A-Knot

Find Knot

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]]

alternate 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 (detailed):

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, [(0, 3, 1, 4), (2, 5, 3, 0), (4, 1, 5, 2)]) is the mirror image of the trefoil knot K3a1, which is a reversible knot and is the first knot in the list with PD code [(0, 3, 1, 4), (2, 5, 3, 0), (4, 1, 5, 2)].

Output format (simple):

We give a string listing the name of each prime summand of the found knot. For instance, 'K3a1#K3a1#K4a1' is the connected sum of two copies of the trefoil and one copy of the figure-eight knot. This does not give all the information included in detailed outputs. In particular, there is no information about symmetry version or symmetry type, so the mirror or reverse of any summand would give the same simplified output.

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.