# What mathematics is behind those metal knot puzzles/brainteasers?

You know those puzzles which are basically two thick twisted rods that look impossible to pull apart. Here is a video,

https://www.youtube.com/watch?v=U5R2GksBqmc

There is a solution where you can pull them apart smoothly, without force. Can you approach the solution to these puzzles mathematically. Is there any theorems/methods you can apply when solving these to help? What field of mathematics explains these puzzles and their solutions? If you can please recommend a book, ressource, lecture notes or course which can teach these fundamentals, or even specifically which goes into the math behind these puzzles. They provide endless hopeless fun!:). Right now I basically approach these by trial and error but i'm hoping I can apply more methodical thinking here. My mind is blank when I get to these lol.

• Not an answer, but: Because you cannot deform, it is not really topology. You may also be interested in the sofa problem Nov 13 '20 at 21:42
• en.wikipedia.org/wiki/… might give you some leads. Nov 13 '20 at 22:12
• Nov 13 '20 at 22:19
• @BarryCipra reminds me of wire helices, placed as in the double helix. Mattresses used to have long wire helixes; one could place two of them around a straight central wire. This is disguised and made unfamiliar as the ends straighten out.. Nov 14 '20 at 1:00
• One of my colleagues picked up a difficult disentanglement puzzle that collective hours had been spent trying to solve it. We did a calculation with fundamental groups (some variation on the Wirtinger presentation) to show a certain assumption about a solution wouldn't make it impossible -- though I can't remember if this led to the eventual solution! Nov 14 '20 at 7:39