Todd Rowland writes: "Consider the number of ways an infinitely stretchable string can be tied around a tree trunk... For any integer n, the string can be wrapped around the tree n times, for positive $n$ clockwise, and negative n counterclockwise. Each integer n corresponds to a homotopy class of maps from $S^1$ to $S^1$."
I do not understand why every $n$ corresponds to a different homotopy class? Why can't we deform a string tied 2 times around the tree, by pulling one of its ends and tying it around the tree a third time?
What is a formal proof that
- For each $n$ there is a single class corresponding to $n$ rounds,
- For different $n$ the classes are different?