The order parameter field u changes as we move around the loop. A loop around the hole corresponds to an extra column of atoms. Moving the atoms slightly will deform the loop, but won't change the number of times the loop winds through or around the hole. Two loops which traverse the torus the same number of times through and around are equivalent. This is why they are called topological defects.
Topology is the study of curves and surfaces where bending and twisting is ignored.
An order parameter field, no matter how contorted, which doesn't wind around the torus can always be smoothly bent and twisted back into a uniform state. In the traditional description, this corresponds precisely to the number of extra rows and columns of atoms we pass by. This was called the Burger's vector in the old days, and nobody needed to learn about tori to understand it. Here's where in the lecture I show the practical importance of topological defects. Unfortunately for you, I can't enclose a soft copper tube for you to play with, the way I do in the lecture.