A knot is a knotted circle embedded in the three-dimensional space as shown in the picture left below. You can see intuitively that the two knots on the right are knotted, and three knots are mutually distinct. However, what would you do if you are asked to prove it mathematically?

In knot theory we associate polynomials (possibly with negative powers) to these knots. If these polynomials are different, then we conclude that these knots are different. The picture on the right shows how to compute the celebrated Jones polynomial. By using the recurrence formula we can compute the Jones polynomials of the three knots on the left, and see that these are indeed different. We often replace intuitive objects such as knots with algebraic objects such as polynomials in knot theory.

By removing a knot itself from the three-space (imagine that we remove a doughnut) and refilling it in another way (twist a doughnut and refill it), we can construct a totally new space. We call such a space a three-manifold. Note that this space is locally the same as the three-space that we live in.