This question was inspired by the very first exercise in Thurston's Three Dimensional Geometry and Topology, where he gives a picture of a very tangled up loop and asks what manifold it depicts.
I understand there are several knot invariants and algorithms that can be used to detect the unknot. But where are the practical implementations? Are there programs into which one can feed an actual picture of a knot (as opposed to more directly computational representations) and have it at least recognize unknots? One test would be whether it would accept the above picture as input and give the correct answer.
It's somewhat limited in its abilities to do so, but I made KnotFolio so that you can paste in a picture of a knot and (with some massaging) turn it into some computational representation and calculate knot invariants.
Starting with a higher-resolution screenshot of the first exercise in Thurston and Levy, "Three-Dimensional Geometry and Topology", pasting it in I get a screen that looks like
If you want, you can lower the "Threshold" slider to thin out the lines a little. I did this, though I didn't need to, and clicked "Accept." This brought me back to the painting mode, if it needs any more touching up.
Clicking "Convert to diagram" does some simple image recognition algorithms and comes up with a combinatorial description. If there are broken or merged lines (from being too thick) it might fail at this step.
We can read off that it has 6 components and 0 crossings, so as a knot this must be a disjoint union of 6 unknots. In particular, to answer the exercise, the manifold is a disjoint union of 6 circles.
We can also take the knot from Thurston's notes (which the book is based on):
With a little bit of editing from inside KnotFolio I can clean it up to look like this:
This is convertable to a diagram:
I only have the KnotInfo data for knots with up to 12 crossings (and the LinkInfo data for links with up to 11 crossings). This is a 38-crossing knot, and it's so big that some of the internal computations get a bit too big for its data types (hence the "assertion failed" error, likely due to a failure to compute the Conway potential). At least the Kauffman bracket shows the knot is nontrivial.
While we're here, one fun feature is that you can click Beautify and (after waiting for an unoptimized algorithm to complete) get to see
Going back to the original diagram, you can also identify the manifold using an image editor. I'm using GIMP here. We know that the only connected 1-manifold with empty boundary is a circle, so the problem reduces to counting connected components.
First I used the bucket tool to create a black/white coloring one way, and I needed to click two times:
Second I used the bucket tool again to create a black/white coloring the other way, and I needed to click five times:
Using some basic algebraic topology (or, less sophisticated, some innermost disk induction argument) one can see that in a black-white coloring of the plane with $B$ black regions, $W$ white regions, and $C$ curves, then $B + W = 1 + C$. With $B=2$ and $W=5$ we get $C=6$.
Hence this manifold is homeomorphic to a disjoint union of six circles.
(In retrospect, one doesn't need to count black and white regions separately, and one can calculate $B + W$ by counting how many clicks it takes to fill in the entire picture.)