Knot theory is a relatively new branch of mathematical research. In the 1880s, Lord Kelvin proposed to model atoms as knots of vertices in the ether. Although this model was not very successful (the existence of ether was disproved in the Michelson-Morley Experiment, laying the foundations for Einstein's special relativity), knot theory remained a vivid area of mathematics research, and its applications e.g. for physics are far from being given up, including brand new areas like quantum computation. Knot theory many areas which are accessible even for undergraduates. From the physical point of view, we construct a knot by taking a segment of rope or a cord, knotting it and fusing the endpoints. A more strict definition from the mathematical point of view is that a knot is a closed loop in three-dimensional space.
To start studying knot theory, you should be already acquainted with the following: