everyone. I have a topological question on the equivalence of loops.
In a two dimensional space, there are some points. These points are fixed and cannot be moved. I have several loops encircling these points. I define two loops are equivalent if they can deform into each other without crossing any points. For example, in the left figure below, Loop 1 is equivalent to Loop 2. And Loop 1 is not equivalent to Loop 3, since they encircle different points and they need to cross points to deform into each other. A more complex example is in the right figure, Loop 4 is not equivalent to Loop 5. Both of them only encircle P1 and P2, but Loop 4 needs to cross P4 twice to deform into Loop 5.
Now I would like to know if there is any formal mathematical language to describe this kind of topology, since my major is physics, not mathematics. Or where I can get more information about this topic.
You're looking for homotopy theory, and in particular for the fundamental group of the punctured plane. The definition of "equivalent loops" that you state is exactly what you have in mind when you state that two loops are homotopic; it means that there is a continuous function $F: I \times I \rightarrow \mathbb R^2 \setminus$ k points such that F(-,0) is the first loop, and F(-,1) gives you the second loop. You can type "Fundamental Group" on Wiki for further reading, and I also recommend you the first chapter of the masterpiece "Algebraic Topology" by Allen Hatcher (it's available on his website).