Knot soliton in DNA and geometric structure of its free-energy density

In general, the geometric structure of DNA is characterized using an elastic rod model. The Landau model provides us a new theory to study the geometric structure of DNA. By using the decomposition of the arc unit in the helical axis of DNA, we find that the free-energy density of DNA is similar to the free-energy density of a two-condensate superconductor. By using the φ-mapping topological current theory, the torus knot soliton hidden in DNA is demonstrated. We show the relation between the geometric structure and free-energy density of DNA and the Frenet equations in differential geometry theory are considered. Therefore, the free-energy density of DNA can be expressed by the curvature and torsion of the helical axis.