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

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

where

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

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 $\beta\rightarrow 0$

$\mathbf{\mathcal{LK}}(K,K')=\mathbf{\mathcal{TW}}(K)+\mathbf{\mathcal{WR}}(K_{h},K_{h})$

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

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

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

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