Way to determine the type of a knot given by a diagram

Question

Is there a general way to determine the type of a knot given by a diagram? I am using KLO (Kirby calculator) and I encounter some nontrivial knots while doing this. For example, can we determine the type of the following diagram?

Answer 1

The best currently known algorithm for unknot recognition is due to Marc Lackenby, it runs in quasi-polynomial time $c^{\log c}$, where $c$ is the crossing number of the knot diagram. There is not (yet) a paper out, but Lackenby gave several talks on this in the last year or so (see Ian Agol's answer here for references). The oldest known algorithm for knot recognition (which has exponential run time) is due to Haken. It is implemented in SnapPy and Regina programs. Lackenby's algorithm is a very clever (and major!) improvement of Haken's idea. However, afaik, it is not yet implemented.

Answer 2

I pasted this diagram into KnotFolio, where it computes a few invariants like the Jones and Alexander polynomials. It has a database of all knots up to 10 or 11 crossings (courtesy of KnotInfo), and, assuming the knot is prime, the only possible knot is $8_1$.

The Alexander polynomial is reportedly $3t^2-7t+3$, which is irreducible, so if the knot were a connect sum one of the summands would have to have Alexander polynomial equal to 1. This is a ten crossing knot, so summands must have 3-7 crossings, and KnotInfo shows there are no such knots, so it is indeed prime.

KnotFolio also outputs the PD code for SnapPy, which we can ask to try to identify by looking for a homeomorphism of the knot exterior that is an isometry taking meridians to meridians (assuming the knot has a hyperbolic exterior). Using SnapPy in Sage:

sage: import snappy
sage: L=snappy.Link([(19,17,20,16), (17,12,18,13), (1,5,2,4),
....: (3,8,4,9), (20,7,1,8), (9,2,10,3), (10,15,11,16),
....: (14,6,15,5), (11,18,12,19), (6,14,7,13)])
sage: L.exterior().identify(True)
[m074(0,0), 8_1(0,0), K5_2(0,0), K8a11(0,0)]

SnapPy is indeed able to solve the gluing equations for a hyperbolic structure on L.exterior() (L.exterior().solution_type() gives "all tetrahedra positively oriented"), so 8_1 it is.

Answer 3

If you have a diagram of a knot with no more than 16 crossings, then I found the following method the quickest.

Save your image of a diagram to a file. Upload it into https://kmill.github.io/knotfolio/ then copy the resulting DT code, and paste it into https://knotinfo.math.indiana.edu/homelinks/knotfinder.php then click Find.