Is it a knot, or not a knot? Reidermeister moves help you resolve such a dilemma. Mathematical knots can be rather deceitful when they're complicated. What looks like a knot may in fact be an unknot. It may be possible to untangle the thing until it becomes a perfect loop, an unknot. It is quite difficult to tell just by visualizing the questionable knot. One way of determining its knotishness is by making a physical model of it and tugging away until you're out of ideas. Another way is to make a 2-dimensional projection of the knot (its shadow) and examine the crossings. When untangling a pseudo-knot, there are three possible moves that could eventually give you an unknot. These moves are called Reidemeister moves. They look somewhat like this:

Move 1:

|*********
|*********
|**/----\**
\*/*****\*
*\******|*
/*\*****/*
|**\----/**
|**********
|**********

to

|.....
|.....
\.....
.\....
..\...
...|..
../...
./....
|.....
|.....

Note that at the end of this move, the string is slightly twisted, but the strings that mathematical knots are made of are 1-dimensional threads and so are infinately stretchy!

Move 2:

|.....|.....
|.....|.....
.\....|.....
..\--|-....
......|..\..
......|...\.
......|...|.
......|../..
../--|-....
./....|.....
/.....|.....
|.....|.....
|.....|.....

to

|...|
|...|
|...|
|...|
|...|
|...|

Note that the curved string is under the straight string, it does not cross over.

Move 3: \.|....../.
.\.\..../..
..\.\../...
...\.\/....
....\/\....
..../\.\...
.../..\/...
../.../\...
./.../..\..
/.../....\.

to

\.|....../.
.\|...../..
..\..../...
..|\../....
..|.\/.....
..|./\.....
..|/..\....
../....\...
./|.....\..
/.|......\.

Note again that the single (eventually) straight string is crossing under the two diagonal strings, so it can be shifted over with ease.

Here comes the best part: Any unknot will at most take 2^(100,000,000,000n) Reidemeister moves to untangle, where n is the number of times the string crosses itself.

Go try it on your own impossibly complicated pseudo-knot today!