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  H has only zero eigenstates and the Hilbert space is finite. For a 3-manifold  \mathbb{M}=\mathbb{R}^{3} the VEV of the Wilson loop is

\langle W[K]\rangle = \int DA\exp(iS_{cs})W[K]   = {\displaystyle \int} \mathcal{D}A\exp \bigg(\frac{ik}{4\pi}{\displaystyle \int_{\mathbb{M}}} tr({A} \wedge  d{A}+\frac{2}{3}{A} \wedge  A \wedge A)\bigg) exp\bigg(i\int A\bigg)


S_{CS}=(ik/4\pi g){\displaystyle \int_{\mathbb{M}}}Tr({A}\wedge d{A}+\frac{2}{3} {A} \wedge A \wedge A\wedge A)

for k \in \mathbb{Z} and g=1, and  A is an  SU(2) connection or gauge field. The partition function Z(\mathbb{M}) 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

  \langle W[K] \rangle  =  \int \mathcal{D}A exp \bigg (\frac{ik}{4\pi}{\displaystyle \int_{\mathbb{M}} }tr({A}\wedge d{A})\bigg) exp\bigg(i {\displaystyle \int} A\bigg)=exp(\mathbf{\mathcal{L}k}[K, K'])

The Gaussian linking number \mathcal{GL}(K,\ \bar{K}) of the curves (K,\bar{K}) describes the number of times one curve intersects the surface bounded by the other.

  \mathcal{GL}(K,\bar{K})= \mathcal{LK} =\frac{1}{4\pi}{\displaystyle \int_{K}}dx^{i} {\displaystyle \int_{\bar{K}}}dy^{j}\epsilon_{ijk}\frac{(x-y)^{k}}{|x-y|^{3}}

and for closed curves or knots  (C,\bar{C}) forming a link

 \mathcal{GL} (K,\bar{K})=\mathcal{LK}\frac{1}{4\pi}{\displaystyle \oint_{\mathfrak{S}}}dx^{i}{\displaystyle \oint_{\bar{C}}}dy^{j}\epsilon_{ijk}\frac{(x-y)^{k}}{|x-y|^{3}}

Equivalently in terms of the paremetrization s \in[0,1],

  \mathcal{GL}(K,\bar{K})=\mathcal{LK}= \mathcal{TW}_{o} =  \frac{1}{4\pi}{\displaystyle \int_{K}} {\displaystyle \int_{\bar{K}}}ds  ds'   [\frac{d\vec{y}(s')}{ds'} \wedge \frac{d\vec{y}(s')}{ds'}]   \frac{(\vec{x}(s)-\vec{y}(s'))^{k}} {|\vec{x}(s)-\vec{y}(s')|^{3}}

Although this is a topological invariant it ceases to be so for the self-linking \mathcal{GL}(K,\bar{K}) for when the curves coincide. However,  \mathcal{GL}(K, \bar{K}) can still always describe the properties of a ribbon with edges  K and \bar{K} that self-entangles or knots about its main helical axis K_{h}

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 \mathcal{J} and \bar{\mathcal{J}} flow in two loops K and \bar{K'} then the ‘action-at-a -distance’ formula for the mutual energy is

 E\sim -\mathcal{J}\bar{\mathcal{J}}\int_{K}\int_{\bar{K}}d\mathbf{r}.d\mathbf{r'} \frac{1}{|\mathbf{r}-\mathbf{r'}|}

which is essentially a knot energy functional. The power of Maxwell’s theory subsequently explained the result in terms of a vector potential \mathbf{A}(\mathbf{r}), with the magnetic field \mathbf{B}(t) essentially the ‘helicity’ \mathbf{B}(\mathbf{r})=\nabla\wedge \mathbf{A}(\mathbf{r}). Suppose  \mathbb{D}\subset\mathbb{R}^{3} and let (K,K')\subset\mathbb{D} be curves or knots within domain \mathbb{D} bounding 2-surfaces \Sigma and \Sigma' such that K=\partial\Sigma and k'=\partial \Sigma'. Let  \mathbf{B}=\mathbf{B}(x) be a magnetic field defined for all x\in\mathbb{D}. The field obeys the Maxwell equations so that

\nabla \wedge \mathbf{B}=\mu_{o}\mathcal{J},~~~\nabla . \mathbf{A}=0  where \mathcal{J} is the current density within the wire. Ampere’s Law follows from Stoke’s Theorem as

  \int_{K}\mathbf{B}.  dl  = {\displaystyle \int_{K}} (\nabla \wedge \mathbf{B}).ds = \mu_{o} {\displaystyle \int_{\Sigma}}   \mathcal{J}. ds=\mu_{o} I 

If the wire pierces the surface \Sigma m times then {\displaystyle \int_{K}} \mathbf{B}. d\mathbf{l}=\mu_{o} m I. If we know introduce the vector potential  \mathbf{A} then  B=\nabla\wedge \mathbf{A} and the Coulomb gauge is  \nabla \mathbf{A}=0 then  B=\nabla \wedge \nabla\wedge A = \nabla^{2}A=\mu_{o}\mathcal{J}. The solution for the vector potential is

  \mathbf{A}(\vec{r})=\frac{\mu_{o}}{4\pi}\int_{\mathbb{D}} d^{3}\mathbf{r'}\frac{J(\vec{r}')}{|\mathbf{r}-\vec{r}'|} = \frac{\mu_{o}}{4\pi}{\displaystyle \int_{K'}} \frac{I d\mathbf{l}'}{|\vec{r}-\vec{r}'|})

since  \mathcal{J}(\mathbf{r}')d^{\mathbf{r}'}=\mathcal{I} d\mathbf{l}' as the current only has support within the wire. The magnetic field is then

 \mathbf{B}(\mathbf{r})=\nabla\wedge\mathbf{A}(\mathbf{r})=\frac{I\mu_{o}}{4\pi}\nabla \wedge {\displaystyle \int_{K'} \frac {d\mathbf{l}}{|\mathbf{r}-\mathbf{r}'|}}  =-\frac{I\mu_{o}}{4\pi}{\displaystyle  \int_{K'}}\frac{(\mathbf{r}-\mathbf{r}^{\prime})\wedge d\mathbf{l}^{\prime}}{|\mathbf{r}-\mathbf{r}^{\prime}|^{3}} 

Introducing a second curve  K and integrating  \mathbf{B} around  K gives

 \int_{K}\mathbf{B}(\mathbf{r}).d\mathbf{l}={\displaystyle \int_{K}}(\nabla\wedge\mathbf{A})(\mathbf{r}).d\mathbf{l}=-\frac{I\mu_{o}}{4\pi} {\displaystyle \int_{K}\int_{K'}}\frac{(\mathbf{r}-\mathbf{r}^{\prime})\wedge d\mathbf{l}^{\prime}.d\mathbf{l}}{|\mathbf{r}-\mathbf{r}^{\prime}|^{3}}=\mu_{o}\pi m 

It can then be seen that

 \mathbf{\mathcal{GL}}(K,K')=m=-\frac{1}{4\pi}{\displaystyle \int_{K}}{\displaystyle \int_{K'}}\frac{(\mathbf{r}-\mathbf{r}^{\prime})\wedge d\mathbf{l}^{\prime}.d\mathbf{l}}{|\mathbf{r}-\mathbf{r}^{\prime}|^{3}}=\frac{1}{4\pi} {\displaystyle \int_{K}\int_{K^{\prime}}} \epsilon_{ijk}dx^{i}dy^{j}\frac{(x-y)^{3}}{|x-y|^{3}} 

The rhs of (-) is the modern expression of the linking number but if \mathbf{r}=(x,y,z) and \mathbf{r}'=(x^{\prime},y^{\prime},z^{\prime}) then (-) it can be expressed as

 m=-\frac{1}{4\pi}{\displaystyle \int\int} \frac{(x-x')(dydz^{\prime}-dzdy^{\prime})+(y^{\prime}-y)(dzdx^{\prime}-dxdz^{\prime}+ (z-z^{\prime})(dxdy^{\prime}-dydx^{\prime})} {|(x^{\prime}-x)^{2}+(y^{\prime}-y)^{2}+(z^{\prime}-z)^{2}|^{3/2}} 

(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, \mathcal{GL}(K,\bar{K}) 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   R(K,\bar{K},K_{h},\beta,\pm \pm n)  be a (twisted) ribbon with edges (K,K') and width \beta, then if  \mathcal{LK} is identified with  \mathcal{GL}(K,K'), then the linking number is the sum of twist and writhe about the helical axis K_{h} such that in the limit as K\rightarrow K_{h} and \bar{K}\rightarrow K_{h} as \beta\rightarrow 0


where the twist is defined as

\ \mathbf{\mathcal{TW}}(K_{h})=\frac{1}{2\pi}\int_{\mathfrak{S}} ds \frac{d\vec{x}(s)}{ds}.\left[\vec{N}(s)\wedge\frac {d\vec{N}(s)}{ds}\right]/[d\vec{x}(s)/ds]

and the writhe is

  \mathcal{WR}(K_{h})  = \frac{1}{4\pi}  {\displaystyle \int_{K_{h}}} ds     {\displaystyle \int_{K_{h}}}ds'    [\frac{d\vec{x}(s)}{ds}  \wedge    \frac{d\vec{y}(s')}{ds'}.\frac{\vec{x}(s)-\vec{y}(s')}{|\vec{x}(s)-\vec{y}(s')|}

where  \vec{N}(x(s))=\vec{N}(s) is the unit vector pointing to  y(s) and   y(s)=x(s)+\beta  \vec{N}(s). If   \vec{N}(s) is the normal vector then   \mathbf{T}_{w} is the total integrated torsion of the single curve  K_{h}

Let  \mathbb{M}^{2+1}=\mathbb{R}^{2}\times\mathbb{R} be a (2+1)-dimensional manifold and consider an Abelian \mathbb{G}=U(1) Chern-Simons gauge field with cpts. A_{\mu}(x)=(A_{o}(x),A_{1}(x),A_{2}(x)). The basic action is

   S_{CS}[A]=\frac{k}{2} \int_{\mathbb{M}} d^{3}x \epsilon^{\mu\nu\rho}A_{\mu}(x)\partial_{\nu} A_{\mu}(x) = \frac{k}{2}{\displaystyle \int_{\mathbb{M}}} A  \wedge  dA 

which is invariant under gauge transformations. This is also analogous to the helicity integrals of (-) except here the integral is over \mathbb{M}^{2+1} and not \mathbb{R}^{3}. One defines a product of Wilson loops

 W[\lbrace q_{i}\rbrace;A]]=\prod_{i=1}^{r} exp\left(iq_{i}\int_{\mathfrak{S}_{i}}A_{\mu}(x)dx^{\mu}\right)\equiv exp\left(iq_{i}\int_{\mathfrak{S}_{i}}A\right) 

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

 \left\langle W[\lbrace q_{i}\rbrace;A]]\right\rangle =\frac{\left\langle\psi_{o}\|W[\lbrace q_{i}\rbrace;A]]\|\psi_{o}\right\rangle}{\left\langle\psi_{o}|\psi_{o}|\right\rangle} = \frac{{\displaystyle\int}\mathcal{D}A W[\lbrace q_{i}\rbrace ;A]]\exp(iS_{CS}[A])}{{\displaystyle\int} \mathcal{D}A exp(iS_{CS}[A])}

where |\psi_{o}\rangle is the vacuum state and \int\mathcal{D}A is the functional integral over the gauge field. The calculation can be done exactly (refs) and is

  \langle W[q_{i} ; A]  \rangle  =  exp\bigg(\frac{i}{2k}   \sum_{i,j} q_{i}  q_{j}  \mathcal{GL} (K_{i},K_{j}) \bigg)  

where   \mathcal{GL}(K_{i}, K_{j})  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    y_{\mu}(s)=x^{\mu}(s)+\beta n^{\mu}(s), then the self intersection is avoided since every curve   K_{i} is smeared out into thin ribbon of width   \beta with edges  K_{i} and  \bar{K}_{i} (ref Witten.) If  q_{i}=q_{j}=Q then

  \bigg( \langle W(K)  \bigg\rangle = exp \bigg(\frac{i q^{2}}{2K}\mathcal{GL}(K,\bar{K})\bigg) 

which describes a ribbon with edges  (K,\bar{K}). 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

  R(K, \bar{K},\beta)  is twisted by   +2\pi then   \mathcal{GL} (K, \bar{K}) \rightarrow \mathcal{GL}(K,\bar{K})+1 . Then

 \langle  W(K) \rangle \rightarrow   exp \bigg(\frac{i q^{2}}{2K}\bigg)\langle  W(K)  \rangle 

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  K within a non-conducting fluid filling a domain  \mathbb{D}. Then a point   x\in K and a point  y 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   K\subset\mathbb{D} \subset\mathbb{R}^{3} be a charged knot and let   \mathcal{D}_{K}(x,y) be the shorter arc length between  x\in K and y\in \bar{K}. The unregulated knot energy functional for K is then


which blows up at  x=y. To regulate   \overline{\mathcal{K}ef}(K,\bar{K}), the following are defined

  V^{j}(K:x,\epsilon)  = \int_{K,  y\in K, \mathcal{D}_{K}(x,y)\ge \epsilon}\frac{dy^{j}}{|x-y|^{2}}

  E(K,\epsilon)=\delta_{ij}\int_{K}V^{j}(K:x,\epsilon)dx^{i}=\delta_{ij} \int_{K} \int_{K,\mathcal{D}_{K}(x,y)\ge  \epsilon}  \frac{dx^{i}dy^{j}}{|x-y|^{2}} 

These have the limits (ref)

V^{j}(K:x,\epsilon)=lim_{\epsilon\rightarrow 0}\bigg(\int_{K, y\in K,\mathcal{D}_{K}(x,y)\ge \epsilon}\frac{dy^{j}}{|x-y|^{2}}-\frac{2}{\epsilon}\bigg) + 4

 E(K,\epsilon)=\delta_{ij} \int_{K}V^{j}(K:x,\epsilon)dx^{i}=\delta_{ij} \int_{K} \int_{K,\mathcal{D}_{K}(x,y)\ge \epsilon} \frac{dx^{i}dy^{j}}{|x-y|^{2}}-\frac{2}{\epsilon}\bigg) + 4

The term  V^{j}(K:x,\epsilon)=\lim_{\epsilon\rightarrow 0}\bigg(\int_{K, y\in K,\mathcal{D}_{K}(x,y)\ge \epsilon}\frac{dy^{j}}{|x-y|^{2}}-\frac{2}{\epsilon}\bigg) + 4 is the “voltage” at x when the subarc  \lbrace y\in K |D_{K}(x,y)\ge \epsilon \rbrace is charged. The regulated knot energy functional of O’Hara is then (ref)

\mathcal{K}ef(K)= E(K,\epsilon)=\delta_{ij}\bigg( \int_{K} \int_{K,\mathcal{D}_{K}(x,y)\ge \epsilon} \frac{dx^{i}dy^{j}}{|x-y|^{2}}-\frac{1}{(\mathcal{D}_{K}(x,y))^{2}}\bigg)dx^{i}dy^{j} 

For open knots, the  +4 term can be dropped and for  K a circle or straight line  \mathcal{K}ef(K)=0 .

Using the electrostatic analogy, knot energy type functionals can be defined for a pair of open or closed curves or knots K,\bar{K})\in\mathbb{D} Let (x,y)\in \mathbb{D} and let (q_{1},q_{2}) be a pair of charges at (x,y). The Coulomb law is then

\mathcal{F}_{q_{1}q_{2}} \sim \frac{1}{4\pi}\frac{q_{1}q_{2}}{|x-y|^{2}}

If the charges are smeared out into strings or curves (K,\bar{K}) then q_{1}=\int_{K}Q_{1}dx and q_{2}=\int_{\bar{K}}Q_{2}dy, where (\mathcal{Q}_{1},\mathcal{Q}_{2}) are charge densities. This gives

\mathcal{F}_{\mathcal{Q}_{1},\mathcal{Q}_{2}} \sim \frac{1}{4\pi}\int_{K}\int_{\bar{K}}\frac{\mathcal{Q}_{1}\mathcal{Q}_{2}}{|x-y|^{2}}dxdy

Setting \mathcal{Q}_{1}=\mathcal{Q}_{2}=1 we can define a non-regulated knot energy functional of the form

  \mathbf{\mathcal{K}ef}(K,\bar{K})=\frac{\delta_{ij}}{4\pi}\int_{K}\int_{\bar{K}}\frac{ dx^{i}dy^{j}}{|x-y|^{2}} 

Introducing a correlation length \zeta, the dimensionless form is

\mathbf{\mathcal{K}ef}(K,\bar{K})=\frac{\delta_{ij}\zeta^{2}}{4\pi}\int_{K}\int_{\bar{K}}\frac{ dx^{i}dy^{j}}{|x-y|^{2}}