A famous result of Witten is a derivation of the Gaussian linking number from an Abelian Chern-Simons theory and the Jones polynomials from the non-abelian theory, thus providing a bridge between quantum theory and knot theory. The Chern-Simons theory is a topological quantum field theory: the Hamiltonian has only zero eigenstates and the Hilbert space is finite. For a 3-manifold the VEV of the Wilson loop is

where

for and g=1, and is an connection or gauge field. The partition function is itself an invariant (the Witten invariant) of the 3-manifold and can be calculated analytically. Witten (refs) demonstrated that this connects with the Jones polynomial of knot theory and the simple non-Abelian CS theory reproduces the Gaussian linking number

The Gaussian linking number of the curves describes the number of times one curve intersects the surface bounded by the other.

and for closed curves or knots forming a link

Equivalently in terms of the paremetrization ,

Although this is a topological invariant it ceases to be so for the self-linking for when the curves coincide. However, can still always describe the properties of a ribbon with edges and that self-entangles or knots about its main helical axis

There is a natural connection between basic knot theory and magnetostatics. Historically, the Gauss Linking Number arose from the Biot-Savart law of magnetostatics, found in 1820. If two currents and flow in two loops and then the ‘action-at-a -distance’ formula for the mutual energy is

which is essentially a knot energy functional. The power of Maxwell’s theory subsequently explained the result in terms of a vector potential , with the magnetic field essentially the ‘helicity’ . Suppose and let be curves or knots within domain bounding 2-surfaces and such that and . Let be a magnetic field defined for all . The field obeys the Maxwell equations so that

where is the current density within the wire. Ampere’s Law follows from Stoke’s Theorem as

If the wire pierces the surface m times then . If we know introduce the vector potential then and the Coulomb gauge is then . The solution for the vector potential is

since as the current only has support within the wire. The magnetic field is then

Introducing a second curve and integrating around gives

It can then be seen that

The rhs of (-) is the modern expression of the linking number but if and then (-) it can be expressed as

(Gauss C. F. 1. Zur mathematischen theorie der electrodynamischen Wirkungen (1833), Werke. K¨oniglichen Gesellschaft der Wissenschaften zu G¨ottingen, 5, 605, (1877).)

This is the original expression first presented by Gauss.

While this historical origin is very far removed from molecular biology, also provides a natural description for a self-linking or self-entangling of a ribbon of DNA exhibiting tertiary structure or superhelicity. Here, the DNA ribbon self-intersects or knots about its own helical axis, leading to writhe and twist. A description of this is accomodated by the Calagareanu-White theorem which follows from the Gauss linking number, and first applied to a description of DNA topology by (refs.) Laboratory observed topological states of DNA are consistent with the C-W theorem

Let be a (twisted) ribbon with edges and width , then if is identified with , then the linking number is the sum of twist and writhe about the helical axis such that in the limit as and as

where the twist is defined as

\

and the writhe is

where is the unit vector pointing to and . If is the normal vector then is the total integrated torsion of the single curve

Let be a (2+1)-dimensional manifold and consider an Abelian Chern-Simons gauge field with cpts. . The basic action is

which is invariant under gauge transformations. This is also analogous to the helicity integrals of (-) except here the integral is over and not . One defines a product of Wilson loops

for a set of curves , the trajectories of particles with charges , with $i=1…n$. Gauge invariance requires that are integers (ref Yang.) The vacuum expectation values of the Wilson loops are given by a path integral

where is the vacuum state and is the functional integral over the gauge field. The calculation can be done exactly (refs) and is

where is the linking number. The non-abelian C-S theory can also be solved and connects with the Jones polynomial for knot theory.

To first order and using a framing such that , then the self intersection is avoided since every curve is smeared out into thin ribbon of width with edges and (ref Witten.) If then

which describes a ribbon with edges . This is useful from the point of view of quantum computing where the ribbon edges can represent worldlines of anyons and braiding is achieved by twisting the ribbon. A twist in the ribbon gives a linking of the worldlines and is registered within the Wilson-loop expected values as a phase shift. For example, if a ribbon

is twisted by then . Then

Again, while this is a powerful representation of the linking number in relation to fluctuating curves, ribbons and knot theory, it is a quantum theory. But for molecular biology–and indeed other applications such as knotted filament

lines/vortices in classical statistical turbulence and polymer fluctuations–it would be useful to have a similar mathematical representation but from a \emph{classical stochastic theory}. This can then represent classically fluctuating polymers and ribbon systems that have an inherent stochastic or random structure and behavior, but which fluctuate or writhe about an average geometry/topology. This is highly relevant to large molecules like DNA, RNA, proteins and other structures within cells or in-vitro solutions, which are essentially immersed in a ‘warm sea’ of smaller molecules which induce a Brownian-motion-type bombardment of the larger structure.

\subsection{Knot energy functionals}

The GLN arose originally from magnetostatics. Another form of knot functional can arise from electrostatics by considering charged knots or strings. Suppose we have a uniformly charged knot within a non-conducting fluid filling a domain . Then a point and a point can experience a Coulomb force. The configuration of the knot may the evolve to decrease its electrostatic energy to a minima. Charged polymers in a solution. Knot energy functionals for a single knot, in the mathematical sense, were considered in relation to an electrostatic anology with a modified Coulomb potential (refs.)

Let be a charged knot and let be the shorter arc length between and . The unregulated knot energy functional for is then

which blows up at . To regulate , the following are defined

These have the limits (ref)

The term is the “voltage” at x when the subarc is charged. The regulated knot energy functional of O’Hara is then (ref)

For open knots, the term can be dropped and for a circle or straight line .

Using the electrostatic analogy, knot energy type functionals can be defined for a pair of open or closed curves or knots Let and let be a pair of charges at . The Coulomb law is then

If the charges are smeared out into strings or curves then and , where are charge densities. This gives

Setting we can define a non-regulated knot energy functional of the form

Introducing a correlation length , the dimensionless form is

## Leave a comment

Comments feed for this article