# What knot is this?

My headphone cables formed this knot:

however I don’t know much about knot theory and cannot tell what it is. In my opinion it isn’t a figure-eight knot and certainly not a trefoil. Since it has $6$ crossings that doesn’t leave many other candidates!

What is this knot? How could one figure it out for similarly simple knots?

Arthur’s answer is completely correct, but for the record I thought I would give a general answer for solving problems of this type using the SnapPy software package. The following procedure can be used to recognize almost any prime knot with a small number of crossings, and takes about 10-15 minutes for a new user.

Step 1. Download and install the SnapPy software from the SnapPy installation page. This is very quick and easy, and works in Mac OS X, Windows, or Linux.

Step 2. Open the software and type:

M = Manifold()


to start the link editor. (Here “manifold” refers to the knot complement.)

Step 3. Draw the shape of the knot. Don’t worry about crossings to start with: just draw a closed polygonal curve that traces the shape of the knot. Here is the shape that I traced:

If you make a mistake, choose “Clear” from the Tools menu to start over.

Step 4. After you draw the shape of the knot, you can click on the crossings with your mouse to change which strand is on top. Here is my version of the OP’s knot:

Step 5. Go to the “Tools” menu and select “Send to SnapPy”. My SnapPy shell now looks like this:

Step 6. Type

M.identify()


The software will give you various descriptions of the manifold, one of which will identify the prime knot using Alexander-Briggs notation. In this case, the output is

[5_1(0,0), K5a2(0,0)]


and the first entry means that it’s the $5_1$ knot.