Terrance tao, the smartest mathematician alive was once asked about the toughest concept in mathematics according to him. "Geometry and Topology", he yelled, no doubt on the face. The mathematician with an IQ of 230 is not very good in Topology ( but he can still kill any normal maths geek I believe ). Now you might be thinking 'How geometry can fear the smartest mathematician alive?' Topology is basically a branch of mathematics that deals with shapes and deformation and transformation of these shapes in order to study the generalized nature of the geometry. According to topology a Cube and a sphere are the same, they are said to be topological invariant. To get an intuition you have to take cube made of clay, now can you transform or deform it into a sphere without tearing any of the clay material. The answer is Yes! you can simply transform it into a sphere just by deforming the edges and vertices in a certain symmetric manner.
Just like that, you can simply say that a mug and donut are topological invariant. It goes as follows:
Now, this seems fun, right? you just have to keep one thing in mind that there should be no tearing of geometry. As you can see that the donut and mug have one hole each. You are not allowed to make another hole, you are not allowed to make an infinity out of a donut. This process is known as diffeomorphism. The problem with topology is that it is very complex to analyze. The above examples are just simple ideas to explain the sense of topology. The dimensions in topology are 'n' in number, in terms of mathematics we can understand what is going on but it is tough to imagine.
There is a branch of topology which is known as Knot theory. It is even crazier to even talk about.
In Knot theory, we are just interested in solving knots in a simple manner. Recently there was a very popular problem of knots which have been solved by a fellow graduate student.
You might be thinking why the hell on earth someone is interested in solving knots. Let me tell you one thing, mathematicians are not bound to make you understand any physical meaning of the situation they are dealing with. It is the work of Scientists to encounter problems using such concepts of mathematics. Just like in this case, Biologists have found a way to solve the problem of the complex entanglement of bacterias using knot theory. Since all the knot theory tells us is how to solve a knot.
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