When knot theory was first developed, its major appeal was in its possible applications to chemistry. Lord Kelvin and chemistry motivated the early development of knot theory in the 1880's. Kelvin hypothesized that a substance called ether was what the entire universe was made of and matter could be explained as knots in the ether; however, we now know that this is not true. This early interest gave knot theory the impetus that it needed in order to become a major field of mathematical study.
The first place for knot theory to find a home was in the biologist's toolkit with applications to DNA. In 1953 James Watson and Francis Crick discovered that the basic genetic material of life on earth took the shape of DNA's double helix from there the possibilities of the marriage of knot theory and Deoxyribonucleic Acid (DNA) were endless. What was also found was that DNA often becomes knotted, making it difficult for DNA to carry out its function. There are enzymes called topoisomerases that can perform topological manipulations on DNA. Scientists let these enzymes act on circular DNA performing actions like those illustrated here so that they can then study the function of the enzymes from the resulting knots in the circular DNA. The circular DNA is used because if open-ended DNA is used, the knots cannot be observed, as the ends are free. This is just one of the exciting applications of knot theory in the world of molecular biology.
Another area for the application of knot theory is statistical mechanics. This application was left undiscovered until very recently. Vaughan Jones discovered the connection when computing a new polynomial invariant for knots. In this field, knots can used to represent systems and thereby increase the ease of the study.
Knot theory is one of the most exciting fields of study in math because of its many applications, examples of other applications are molecular chemistry and particle physics, as well as everyday applications the are wont to arise in the future.