The initial interest in knot theory was stimulated by Kelvin's theory of atomic structure. By the turn of the century, after Mendeleev's periodic tables, it was clear that Kelvin's theory was incorrect. Chemists were no longer interested in classifying knots. However, topologists continued to study knots. The focus of chemists turned towards attempts to synthesize molecular KLs. The first pair of linked rings in a form of the Hopf link, a catenane, was synthesized by Frisch and Wasserman in 1961. The first molecular knot, a trefoil made out of 124 atoms was produced by C. DietrichBushecker and J.P. Sauvage in 1989. They refer to stereochemical topology, synthesis, characterization, and analysis of topologically interesting molecular structures. After the synthesis of first molecular Möbius ladder with three rungs by D. Walba, R. Richards and R.C. Haltiwanger in 1982, with the addition of twists to the Möbius ladders managed by Q.Y. Zheng in 1990, numerous KLs become possible. In fact, after breaking the rungs, Möbius multistranded twisted ladders are a molecular closed braid representation of a KL. In the 1950s F.H.C. Crick and J.D. Watson unravelled the double helix structure of DNA. A molecule of DNA can also take the form of a ring and become knotted. In the process of recombination, a DNA knot can be temporarily broken, physically changed, and then reconnected. In the 1970s it was discovered that an enzyme, topoisomerase, is responsible for the complete process from an initial break to the recombination. The first electron microscope pictures of knotted DNA were produced in 1985 (Wasserman, Dungan and Cozzarelli, 1985). The linking number and its splitting into average writhe Wr and twist Tw is used as a basic tool to analyze the geometry of supercoiled DNA. C. Ernst and D.W. Sumners (1990, 1999) reconstructed the actions of enzyme (TN3 Resolvase) by solving tangle equations. Distances of rational knots and links were calculated by I.K. Darcy and D.W. Sumners (2000). V. Jones established a connection between knot theory and statistic mechanics as a part of his discovery of Jones polynomial. Mathematical modelling of large systems of particles (e.g., of phase transitions or magnetization) described by the Ising model, YangBaxter equation, and Potts model, resulted in knot invariants generated by a partition function. Different applications of knot theory in physics, chemistry, and biology are considered in books by L. Kauffman (1991), C. Adams (1994), K. Murasugi (1996), E. Flapan (2000), and in the collection of papers edited by D.W.Sumners (1993). In this book we shall only describe a few applications of knot theory not completely covered by other books, such as the applications of knot theory to ornamental art, and in logic.
