When I first heard about this theorem, it was so fascinating to me that it was one of the first that came to my mind when deciding what I wanted to include on this site. Bear with me, because it is a confusing concept, and I'm still wrapping my head around it. Let's first start by defining a link. I know, the theorem is rational TANGLES, but hear me out. A link is a part of knot theory, which is the study of mathematical knots. This is a section in the field of topology and differ from the typical knots you see in ropes and shoelaces. They are 3 dimensional like knots we see in our day-to-day, however, they are theoretical in the way that the two ends are attached to one another permanently. We're able to define two knots as equivalent if one can be formed to look just like the other. The series of alterations of the one knot to get to the other is known as ambient isotopies. So, links exist in the scope of knot theory. It is defined as an assembly of knots with mutual entanglements. Knots can be considerd links; links that only have one component are considered just knots.

With some generics out of the way, now I think we can define a tangle. So, remember that a link's ends are permanently attached to each other? Well a tangle is just a link with free ends, known as strands. It can get a little complicated, because these strands can be knotted and linked as well. Now, there's this theoretical "box", where the strands enter, but it contains no free ends itself. That's all outside the box. You should be able to begin at one strand and follow it through the box and eventually leave the box to reach the other end of the strand. So, there must always be an even number of strand ends.

The concept of rational tangles was brought upon by John Conway as a foundation for constructing knots, where knots are the simplest form of tangles as they can be "unwound". Rational tangles are formed through picking two endpoints and twisting them, picking another pair and twisting them, again and again for a finite number of times. A horizontal integer tangle is a twist of two horizontal strands a number of times in the positive or negative direction based on the number of times. A vertical integer tangle can be defined the same way, just opposite.