From wikipedia:
A crucial difference between the standard mathematical and conventional notions of a knot is that mathematical knots are closed—there are no ends to tie or untie on a mathematical knot. Physical properties such as friction and thickness also do not apply
What was the need of having this difference? Would it not be better to have mathematical knots much similar to actual phyical knots, so we can study them better?
For a mathemtical theory of knots, you need some notion of when two knots are equivalent. It seems reasonable to say that two knots are equivalent if you can get from one to the other by an "ambient isotopy" (I won't explain what that is, but encourage you to look it up and see that it really is very reasonable to say two knots are the same if you can get from one to the other by an ambient isotopy). Now if you don't tie the ends together, there's an ambient isotopy that just pulls one end through the knot, thus untying it. Thus, the whole theory never gets off the ground, if you don't tie the ends together --- everything is unknotted!
