Anybody who can tie their shoelaces knows about knots! Still, the mathematical theory of knots is interesting. The theory developed both from the study of electricity and early atomic physics. It draws from geometry, group theory, linear algebra (which you can read about at the Mathematical Atlas site) , number theory, algebraic geometry and differential geometry.
Knots come in many different flavors. How many "different" kinds can you think of?
There is the the simple overhand knot
and the figure-eight knot
Find yourself a piece of string, or a shoelace, and tie a simple knot. Is it possible to go from this simple knot to a figure-eight knot without untying or tying the string?
We're talking mathematics here -- so it doesn't matter what you do; no amount of manipulation will prove that these knots are different.
Knot theory is the rigorous mathematics which can distinguish between different knots.
Of course, mathematics is really just a glorified game (check out the poster called What is Mathematical Truth?). Tying knots in a piece of string helped shape the mathematical definition you're about to see, but you should remember that a mathematical knot has nothing to do with your shoelaces! Well, maybe a little something. Mathematical knots are an imagined model of the physical knot-making process. (What properties do you think a good model should have?) This is a common theme in mathematics: a physical situation gives rise to a mathematical model; the theoretical worth of the model is judged by how much insight it provides on the original physical situation.
So what is a mathematical knot?
Well, the first thing to worry about are all those loose ends -- literally. What to do about the ends of the piece of string? Do they matter? How can you keep track of them? Think about this, and then check out the official definition of a mathematical knot.
Now that we know what a mathematical knot is, the next question is: What do we mean when we say two knots are mathematically the same? Is it possible for two knots to be topologically different? After you've thought this through, check out my comments on the mathematical way of telling knots apart.