Statistical topology of closed curves: Some applications in polymer physics

Topological entanglement in polymers and biopolymers is a topic that involves different fields of science such as chemistry, biology, physics, and mathematics. One of the main issues in this topic is to understand how the entanglement complexity can depend on factors such as the degree of polymerization, the quality of the solvent, and the temperature or the degree of confinement of the macromolecule. In this respect a statistical approach to the problem is natural and in the last few years there has been a lot of work on the study of the entanglement complexity of polymers within the statistical mechanics framework. A review on this topic is given here stressing the main results obtained and describing the tools most used with this approach.

Figure 1

Knot diagrams for some knots.

Figure 2

The three Reidemeister moves.

Figure 3

(Color online) Oriented knot diagram for the knot $4_1$. The numbers in boxes refer to the labeling of the four arcs. For clarity the arcs are delimited by filled circles.

Figure 4

The sign convention for signed crossings and the arc labelings for computation of the Alexander polynomial.

Figure 5

(Color online) Oriented knot diagram for the knot $-3_1$.

Figure 6

Two drawings of a punctured torus whose boundary curve is the unknot.

Figure 7

The Seifert surface of the figure-eight knot. This surface is a punctured torus and the figure-eight knot has genus one.

Figure 8

(Color online) The shortest trefoil knot on a cubic lattice.

Figure 9

A sketch of the knotted ball pair corresponding to the trefoil.

Figure 10

(Color online) Some link diagrams. The links shown are the Hopf link, the $(2,4)$-torus link, the Whitehead link, and the connect sum of the trefoil and the Hopf link.

Figure 11

Some orientable ribbons. In the sketch on the left the two boundary curves form the Hopf link. On the right they are a satellite link of the trefoil.

Figure 12

Some nonorientable ribbons. The left figure shows a ribbon whose boundary curve is the trefoil, i.e., the $(2,3)$-torus knot. The right is a satellite knot of the trefoil.

Figure 13

(Color online) Examples of pivot moves: I, inversion. II, reflection through the $x=-y$ line.

Figure 14

Scaling of the unknotting probability for a ring confined to a sphere.

Figure 15

(Color online) Knotting probability as a function of the density of right angles $\rho$. The dashed line represents the unweighted polygons.

Figure 16

Schematic representation of the projection of two chains properly embedded in a cube, and the completion of the chains to form two polygons by adding edges outside the cube.

Figure 17

The BFACF moves.

Figure 18

Localization of a knot in the interior of a sphere $S$.

Figure 19

An example of a flat knot configuration on the square diagonal lattice with six crossings and the topology of a trefoil knot.

Figure 20

A typical configuration for the composite knot $3_1\#3_1$. We have indicated the location of the two prime knots.