Hopefully you noticed that all knots are topologically the same. Topologically, all knots are just circles -- and so they must be topologically identical.

The property that defines any given knot has to do with how it is embedded -- how it sits -- in three-dimensional space.

We say that two knots are the same if we can find a homeomorphism from three-dimensional space to three-dimensional space which maps the one knot onto the other.

This defintion is the right way to go, because it is precisely tearing (or tying and untying) that we want to prevent -- and this is certainly a topological consideration, but we have to take great care in preserving the embedded nature of any given knot.

Of course, in practise, it can be very hard to tell if two knots are different. Take for example, the granny knot and the square knot. See if you can turn one into the other using a piece of string. Remember, you can't tie or untie anything.

the granny knot

and the square knot

After you've played around with these two knots, check my comments.

Mathematicians use algebraic tools to tell knots apart. They attach different algebraic objects -- usually some kind of polynomial -- to knots, and then use the techniques of algebraic topology. You can read much more about this by checking out some further knot links.