Knot Theory is one of the most fascinating areas of Mathematics. Its origins date back to the late nineteenth century and it is currently part of the field called Algebraic Topology. It is striking that some concepts, proofs and open problems can be stated in simple language, allowing non mathematicians to have access to vast portions of the theory.

Prime knots (i.e. non composed) with nine crossings are exactly the ones on the table above. There are 165 different knots with ten crossings!
Image: A Knot Zoo.

Knot Theory studies closed curves in space without self-intersections. Two of these curves (or knots) are considered equivalent if one can be turned into the other by a continuous deformation. In the deformation process, self-intersections, ruptures or collapses (such as a knot so tight that it disappears) are not allowed.

One is usually not concerned with the exact shape of the curves. For example, any curve that can be deformed into a circle can be called a trivial knot.

Besides that, you can also relate some of the knots with its corresponding surfaces, which conects knots and link's study with surfaces topology's. This relation gets very clear in the next video!

If you want to know more about surface topology, check our page that talks about it!

Video: Rodrigo Tetsuo Argenton.

Mathematically, a knot is an insertion of a circle in three-dimensional Euclidean space, the R3. This Wikipedia page explains in detail what these knots are, gives several examples, separates them into categories - such as tame and wild knots - presents their applications in graph theory and shows generalizations that refer to non-traditional knots.

The knot theory itself, so studied by many mathematicians, is explained on this Wikipedia page, which shows its history, explains what equivalent knots are and also tabulates the knots using some notations, such as Alexander-Briggs's and Dowker's. In addition, it brings a brief explanation of the placement of knots in higher dimensions, such as R4!

Still talking about the knot theory, this video from the Abby Geigerman channel intuitively brings a little about this theory, using schematizations and drawings that facilitate its understanding. She talks about some specific knots such as the trefoil and also about Reidemeister's moves, one of three local moves in a link diagram!                            

The knot theory is inserted in the field of Topology, better explained in this Wikipedia page, which talks about its motivation, history, concepts, topics and some applications in different areas of knowledge. More specifically, such a theory is material for studying algebraic topology, detailed on this other Wikipedia page, which shows its main branches, such as homotopy groups.