# Exploring the electric field around a loop of static charge:
Rectangles, stadiums, ellipses, and knots

Max Lipton<EMAIL_ADDRESS>Mathematics Department, Cornell University,
Ithaca, NY 14853  Alex Townsend<EMAIL_ADDRESS>Mathematics Department,
Cornell University, Ithaca, NY 14853  Steven H. Strogatz<EMAIL_ADDRESS>Mathematics Department, Cornell University, Ithaca, NY 14853

###### Abstract

We study the electric field around a continuous one-dimensional loop of static
charge, under the assumption that the charge is distributed uniformly along
the loop. For rectangular or stadium-shaped loops in the plane, we find that
the electric field can undergo a symmetry-breaking pitchfork bifurcation as
the loop is elongated; the field can have either one or three zeros, depending
on the loop’s aspect ratio. For knotted charge distributions in three-
dimensional space, we compute the electric field numerically and compare our
results to previously published theoretical bounds on the number of
equilibrium points around charged knots. Our computations reveal that the
previous bounds are far from sharp. The numerics also suggest conjectures for
the actual minimum number of equilibrium points for all charged knots with
five or fewer crossings. In addition, we provide the first images of the
equipotential surfaces around charged knots, and visualize their topological
transitions as the level of the potential is varied.

††preprint: APS/123-QED

##  I Introduction

In a first course on electricity and magnetism, students are often asked to
solve problems in electrostatics [1]. For example, calculating the electric
field around an infinite line of uniformly distributed static charge provides
good practice in working with Coulomb’s law or Gauss’s law. Another classic
exercise is to calculate the electric field at all points on the symmetry axis
above the center of a uniformly charged circular ring.

Here we explore the electrostatics of one-dimensional charge distributions
that are less standard than lines and circles. First, working with charged
loops confined to a plane, we show that charged rectangles and stadiums can
give rise to symmetry-breaking bifurcations in their surrounding electric
fields. Specifically, imagine stretching a rectangular charge distribution
along its major axis. When the charged rectangle becomes sufficiently
elongated, the electric field it generates can have three equilibria: one at
the center of the rectangle, and another two located on the major axis,
symmetrically placed on opposite sides of the center. These two additional
equilibria emerge from a supercritical pitchfork bifurcation at a critical
aspect ratio that can be calculated explicitly. Surprisingly, nothing like
this happens for a charged ellipse; at all aspect ratios, its surrounding
electric field has only one equilibrium point at its center.

Next we allow our charged loops to move out of the plane and into three
dimensions. In particular we consider charged trefoil knots, figure-eight
knots, and other charged loops that have knots tied in them. Charged knots may
sound at first like a contrived thing to study, but they actually arise quite
commonly in nature; they are found in long closed polymers and specifically in
long loops of DNA [2, 3, 4, 5]. Many new questions about equilibrium points
and equipotential surfaces arise in the context of charged knots that we hope
will appeal to the general physics community.

Before we proceed with the analysis, let us clarify how our work differs from
that recently presented elsewhere by the first author [6, 7]. The articles [6,
7] are aimed at specialists in topology, geometry, and knot theory, whereas
the present treatment is intended for physicists. We are also more concerned
here with giving physical, visual, and numerical results and examples, whereas
the earlier articles concentrated on the precise statements and proofs of
certain theorems we cite below. For example, it was proven in Ref. [7] that
the electric field around any uniformly charged knot must have at least $2t+1$
equilibrium points, where $t$ is a topological invariant known as the knot’s
tunnel number. We now show numerically that this lower bound is not sharp, and
we offer conjectures for the actual minimum number of equilibrium points for
each type of knot with five or fewer crossings. Using computer graphics, we
also visualize the equipotential surfaces of charged knots for the first time.
Furthermore, our results about charged rectangles, stadiums, and ellipses are
new.

##  II Charged Loops in the Plane: Rectangles, Stadiums, and Ellipses

###  II.1 Rectangle

As a first example, consider a one-dimensional loop of static charge in the
$xy$ plane, uniformly and continuously distributed in the shape of a rectangle
with aspect ratio $a\geq 1$. As shown in Figure 1, the rectangle consists of
two horizontal line segments extending from $x=-a$ to $a$, lying one unit
above and below the $x$ axis, capped off by vertical lines at both ends.

Figure 1: A uniform charge distribution in the shape of a rectangle. Static
charge is confined to this one-dimensional loop, and we seek to understand the
electric field that surrounds it in three dimensions.

One of the main topics of interest to us is the location of the zeros of the
electric field in the three-dimensional space around the rectangle. These
zeros correspond to equilibrium points where a test charge could remain at
rest (though not stably, as a well-known theorem forbids the existence of
stable equilibria in an electrostatic field [1]). By symmetry, there must be a
zero at the center of the rectangle. But can there be any other zeros?

Clearly there cannot be any zeros above or below the $xy$ plane, because all
the charges in the rectangle would attract or repel a test charge in the same
direction relative to the $z$-axis. So it suffices to look for zeros in the
$xy$ plane. These correspond to critical points of the potential, where the
gradient vanishes. To find such zeros, if they exist, let us calculate the
potential on the $xy$ plane generated by the rectangular charge distribution.

Consider the contribution to the electric potential from the top side of the
rectangle. Assume a linear charge density of unit strength, for simplicity. As
shown in Figure 1, the distance between a typical line element $dx$ at $(u,v)$
on the top and a point $(x,y)$ inside the rectangle is
$\sqrt{(u-x)^{2}+(1-y)^{2}}$, since $v=1$ on the top of the rectangle. For the
inverse-square electrical force corresponding to Coulomb’s law in three
dimensions, the contribution to the potential is inversely proportional to
this distance. Hence, by integrating over all the line elements on the top of
the rectangle, we obtain a contribution to the potential of

$\int_{-a}^{a}\frac{dx}{\sqrt{(u-x)^{2}+(1-y)^{2}}}.$

This integral evaluates to

$\begin{split}\frac{1}{2}\log\left(\frac{a-x+\sqrt{(x-a)^{2}+(y-1)^{2}}}{a-x-\sqrt{(x-a)^{2}+(y-1)^{2}}}\right)\\\
+\frac{1}{2}\log\left(\frac{a+x+\sqrt{(x-a)^{2}+(y-1)^{2}}}{a+x-\sqrt{(x-a)^{2}+(y-1)^{2}}}\right).\end{split}$
(1)

The contributions to the potential from the other three sides of the rectangle
can be evaluated similarly. In this way one can obtain the potential
everywhere inside the rectangle in the $xy$ plane. (We do not show the final
result because it is unpleasant to look at, but it is a sum of four terms like
that above.)

Figure 2 shows a contour map of the potential $\phi(x,y,z)$ restricted to the
plane $z=0$ for two values of the aspect ratio $a$. When $a$ is close to 1 and
the rectangle is almost square, the only zero is at the origin (Figure 2(a)).
However, if we increase $a$ to 2.5, the rectangle becomes more elongated and
we now find three zeros: one at the origin, and a symmetric pair on either
side of the origin (Figure 2(b)).

Figure 2: Contour map of the electrostatic potential inside a uniform
rectangular charge distribution. (a) Aspect ratio $a=1.1$. The only critical
point (also known as a zero or an equilibrium point of the electric field)
occurs at the center of the rectangle. (b) Aspect ratio $a=2.5$. There are now
three zeros.

To find the threshold value of $a$ at which the bifurcation occurs, we examine
how the potential varies along the $x$-axis. Figure 3 shows that the potential
on the $x$-axis changes from having a minimum to a maximum at $x=0$ as $a$
increases. At the bifurcation value of $a$, the second derivative of the
potential at the origin vanishes as the graph of the potential changes from
concave up to concave down. By calculating this second derivative
analytically, we find that the threshold value of $a$ satisfies

$4+8a^{2}-4a^{3}=0,$

whose unique positive root is $a\approx 2.20557.$

Figure 3: The potential of charged rectangles along the $x$-axis. (a) Aspect
ratio $a=1.1$. The only critical point lies at the origin. (b) Aspect ratio
$a=2.5$. The three critical points where the slope vanishes correspond to
electrostatic equilibrium points, i.e., zeros of the electric field.

###  II.2 Stadium

Our second example takes the form of a planar curve shaped like a stadium
(Figure 4). This curve consists of two equal parallel line segments that run
from $x=-a$ to $x=a$, with $y$ values given by $y=\pm 1$. These line segments
are capped off by semicircles of unit radius at either end. The resulting
stadium curve is well known in studies of chaotic billiards in classical
mechanics [8] and in experimental studies of quantum chaos [9].

Figure 4: A stadium-shaped charge distribution. The stadium’s aspect ratio is
defined as $a+1$, where $a$ denotes the half-length of the straight portion of
the stadium.

We now show that the electrostatic field around a uniformly charged stadium
curve can have either one or three zeros, depending on the value of the
stadium’s aspect ratio. As before, imagine distributing static charge
uniformly and continuously along this curve in the $xy$ plane. Zeros of the
electric field around it must lie in the $xy$ plane, by the reasoning given
earlier for the rectangle. Indeed, the zeros are all confined to the $x$-axis,
for all values of $a>0$.

Figure 5 and Figure 6 show the behavior of the potential for two values of
$a$. For $a=1$, there is only one zero, whereas for $a=2$ there are three
zeros.

Figure 5: (a) Contour map of the electrostatic potential for a charged stadium
with $a=1$. The only critical point lies at the center. (b) Contour map for
$a=2$. Now there are three critical points: one at the center, and a pair on
either side. Figure 6: (a) Potential along $x$-axis for stadium with $a=1$.
(b) Potential along $x$-axis for stadium with $a=2$. The three critical points
where the slope vanishes correspond to electrostatic equilibrium points, i.e.,
zeros of the electric field.

To find the threshold value of $a$ at which the bifurcation occurs, we follow
the same procedure as for the rectangle. The integrals arising in the
expression for the potential along the $x$-axis can again be found
analytically, yielding a formula in terms of elliptic functions and inverse
$\sinh$ functions (we omit the details). By solving for the value of $a$ at
which the second derivative of the potential along the $x$-axis vanishes at
the origin, we find the bifurcation occurs at $a\approx 1.1313$. The
corresponding aspect ratio of $a+1\approx 2.13$ is not too different from the
value of $2.2$ found earlier for the rectangle.

###  II.3 Ellipse

Curiously, when we redo the calculations above for an ellipse, we always find
only a single zero (the trivial equilibrium point at the center expected by
symmetry), no matter what aspect ratio we choose. We have not proven this, but
we find numerically that the second derivative of the potential (restricted to
the $x$-axis) is strictly positive for all aspect ratios and hence all
possible elliptical shapes (Figure 7).

An intuitive explanation of this result relies on the shape difference between
elongated rectangles and stadiums on the one hand, and elongated ellipses on
the other. The key is that there is enough charge on the short side of a
rectangle (or the small semicircle of a stadium) to counterbalance the forces
exerted by the rest of the shape, as long as a test charge is situated
sufficiently close to the short side (or the small semicircle). For example,
consider a highly elongated rectangle. It seems plausible that there should be
an equilibrium point very close to either of the short sides of the rectangle;
all the charge on that side can exert enough electrical force to
counterbalance the weaker forces exerted by the much more distant charges
located on the rest of the rectangle. The same effect holds for a stadium. But
with an ellipse, there is apparently not enough charge on the narrow end of
the ellipse to offset the force coming from the rest of the shape.

\begin{overpic}[width=99.73074pt]{Images/SecondDerivativeEllipse.png}
\put(50.0,-5.0){$a$} \end{overpic} Figure 7: The second derivative in $x$ of
$\phi$ at the origin for a charged ellipse with aspect ratio $a$. Here $a$ is
defined as the length of the ellipse’s major axis, measured in units of its
minor axis. Numerically, it appears that the second derivative of the
potential is strictly positive at the origin. This observation provides
numerical evidence that the electric field associated with any charged ellipse
has exactly one zero.

##  III  Charged Knots

In the rest of this article we move out of the plane and consider a continuous
charge distribution in the shape of a knotted loop in three-dimensional space.
Generalizing our work in the previous section, we ask: How many equilibrium
points must exist, and where are they located in relation to the knot? What do
the equipotential surfaces look like, and how does their topology change as we
vary the level of the potential?

###  III.1  Background

The mathematical theory of knots and links grew out of two problems in
classical physics [10]. In 1833, while studying electromagnetism and mutual
inductance, Gauss discovered a formula for the linking number of two closed
curves in three-dimensional space [11]. A few decades later, Lord Kelvin
proposed his vortex model of the atom, in which different elements were
imagined to arise from different patterns of knotted and linked vortex rings
in the ether [12]. Since then, knots and links have been studied in many other
parts of physics [13, 14, 15, 16], including quantum field theory [17, 18],
liquid crystals [19], plasmas [20, 21], Bose–Einstein condensates [22, 23],
fluids [24, 25], superfluids [26], the biophysics of DNA and other polymers
[2, 4], the mechanics of ropes and elastic rods [27, 28, 29, 30, 31],
classical field theory [32], knotted scroll waves in excitable media [33, 34],
$n$-body choreographies in celestial mechanics [35], and electromagnetic waves
and fields [36, 37, 38, 39, 40, 41, 42, 43].

###  III.2  Physical Intuition about Equipotential Surfaces and Equilibrium
Points

To get an intuitive feel for knotted charge distributions, let us start with
the “trivial knot” (also known as the “unknot”), and for simplicity, consider
its most symmetrical realization: a perfectly circular and uniform loop of
static charge. By symmetry, or by using Coulomb’s law, one can prove there is
an equilibrium point at the center of the circle. Moreover, the symmetry of
the situation suggests that the equipotential surfaces enclosing the circle
should be nested tori, at least sufficiently close to the circle. Far from the
circle, they must approach spheres, because a circular loop of charge looks
like a point source in the far field.

Figure 8: Topologically different equipotential surfaces for a charged
trefoil. Only “regular” values of the potential are shown, at which the
equipotential surfaces are smooth manifolds. Bifurcations (not shown) lie in
between these examples. In the leftmost panel, the potential is very large and
positive close to the knot, and the corresponding equipotential surface
resembles a thin tube around the knot. As we lower the potential
(corresponding to moving from left to right in the figure), the surface
inflates and swells up. At bifurcation values, it self-intersects and creates
new zeros in the electric field, corresponding to new equilibrium points. The
equipotential surfaces shown here have genus values of $1,4,3,$ and $0$,
moving from left to right. Figure 9: A selection of the topologically
different equipotential surfaces for a charged figure-eight knot.

Remarkably, a similar line of reasoning works even if the charged loop has
knots tied in it. Although it becomes impossible to calculate the location of
the equilibrium points analytically in these cases, we can still prove that
equilibria must exist in the surrounding electric field.

To do so, picture the equipotential surfaces. Figure 8 and Figure 9 show what
they look like for a trefoil knot and a figure-eight knot, but the same idea
works for any sufficiently smooth knot. One expects that close to the knot,
the equipotential surfaces must be tubular versions of the knot itself—in
other words, they must be knotted tori—whereas far from the knot, they must
resemble spheres (because a charged knot, like any other compact charge
distribution, appears point-like in the far field). So if we imagine
continuously varying the potential from high levels near the knot to low
levels at infinity, the equipotential surfaces must continuously deform from
knotted tori into spheres. To make this transition, the knotted tori swell up,
collide with distant parts of themselves, and reconnect in ways that alter
their topology. At such collisions, two patches of a single equipotential
surface intersect tangentially. Because the electric field vector lies along
the normal to each of the colliding patches, and because those normal vectors
point in opposite directions at the point of collision, the electric field
must vanish there.

This argument suggests that there must be one or more zeros in the electric
field around any charged knot. Each zero represents an equilibrium point where
a test charge could remain motionless. As is well known, there are no _stable_
equilibria in an electrostatic field, so all these equilibrium points must be
unstable [1]. Indeed, they are all saddle points, with either one- or two-
dimensional unstable manifolds.

These intuitive ideas have recently been sharpened and made rigorous with the
help of Morse theory, algebraic topology, and geometric topology [6, 7]. In
particular, one theorem [7] provides a lower bound on the number of
equilibrium points around any charged knot. It states that the electric field
must have at least $2t+1$ zeros, where $t$ is a topological invariant known as
the tunnel number of the knot [44, 45, 46].

But as we will show below, this $2t+1$ lower bound turns out to be rather
loose. For example, the tunnel number for a trefoil knot is known to be 1
(indeed, the tunnel number is 1 for all prime knots with seven or fewer
crossings [45, 46]), so the $2t+1$ bound implies that a charged trefoil must
always have at least three zeros in its surrounding electric field. Yet we
have never seen fewer than seven zeros in our computations, no matter how the
trefoil is bent, twisted, or otherwise deformed. So is seven the absolute
minimum? Or could there be some needle-in-a-haystack conformation of a charged
trefoil that has three, four, five, or six zeros? The problem is currently
unsolved, and is just one of the many open problems about charged knots.

In what follows, we propose conjectures for the minimum number of zeros for
all prime knots with up to five crossings, based on our numerical experiments.
But proving (or disproving) these conjectures—and extending them to a much
wider range of knots—remains a challenge.

One may reasonably ask why we are so concerned with these zeros, given that
they represent unstable equilibria and are therefore of questionable physical
significance. Although it remains to be seen whether they are physically
important, from a mathematical standpoint they are definitely interesting for
the following reason. The minimum number of zeros for the electric field
produced a charged knot of a given knot type provides a new knot invariant,
and one based in elementary physics. It may be a difficult invariant to
calculate at this early stage. But we are intrigued by the possibility that
subsequent research will find connections between it and other, better
established knot invariants.

Our simulations also allow us to investigate how the equipotential surfaces
change their topology as we vary the level of the potential. This sequence of
topological transitions can be characterized with a set of integers we call
the Morse code for the knot [6]. We compute these transitions numerically for
some convenient embeddings of the simplest knots and illustrate them with
computer graphics (Figure 8 and Figure 9).

###  III.3  Numerical Methods

To describe our computations, we introduce some notation and terminology. Let
the knot $K$ be parametrized by a vector-valued function $\pmb{r}(t)$, where
$0\leq t\leq 2\pi$. Because the knot forms a closed loop, we also require that
$\pmb{r}(0)=\pmb{r}(2\pi)$. Then, from Coulomb’s law, the electric potential
$\phi$ at a point $\pmb{x}\in\mathbb{R}^{3}$ away from the knot is given in
dimensionless form by

$\phi(\pmb{x})=\int_{\pmb{r}\in
K}\frac{|\pmb{dr}|}{|\pmb{x}-\pmb{r}|}=\int_{0}^{2\pi}\frac{|\pmb{r}^{\prime}(t)|dt}{|\pmb{x}-\pmb{r}(t)|},$
(2)

where $|\cdot|$ denotes the magnitude of a vector quantity. (We have written
the potential in dimensionless form for convenience; one could include
physical parameters like the vacuum permittivity $\epsilon_{0}$ or the uniform
charge density $\rho$ along the knot, but we have chosen not to do so as they
play no role in our analysis.) The electric field associated with the
potential is given by $\pmb{E}(\pmb{x})=-\nabla\phi(\pmb{x})$. The zeros
(i.e., the equilibrium points) of the electric field are equivalent to the
critical points of the electric potential $\phi$; as such, we will continue to
use the terms zeros, equilibrium points, and critical points interchangeably,
as we did earlier in Section II.

For most knots, the potential $\phi(\pmb{x})$ and its critical points cannot
be calculated analytically. We must rely on numerical integration and
rootfinding techniques. To perform these computations, we replace the
continuous knot by $N+1$ unit point charges located at
$\pmb{r}(t_{0}),\ldots,\pmb{r}(t_{N})$, where $t_{k}=2\pi k/(N+1)$, and use
the following trapezoidal approximation of (2):

$\phi(\pmb{x})\approx\frac{2\pi}{N+1}\sum_{j=0}^{N}\frac{|\pmb{r}^{\prime}(t_{j})|}{|\pmb{x}-\pmb{r}(t_{j})|}.$
(3)

We use the same trapezoidal approximation for the electric field
$\pmb{E}(\pmb{x}).$ (There are more efficient approaches for evaluating
$\phi(\pmb{x})$ and $\pmb{E}(\pmb{x})$ when $N$ is very large; these are based
on multipole expansions such as those used in the fast multipole method [47].)

As mentioned above, we are interested in finding the zeros of the electric
field. These are defined as points $\pmb{x^{*}}\in\mathbb{R}^{3}$ such that
$\pmb{E}(\pmb{x^{*}})=\pmb{0}$. To find them, we start with initial guesses
and then refine the guesses iteratively, using a multivariable Newton method.
To obtain reasonable initial guesses, we use an algorithm known as “3D
Marching Cubes,” a computer graphics algorithm for finding level sets of a
scalar function [48]. Here, we use Marching Cubes on each of the three
components of $\pmb{E}(\pmb{x})$ to find their zero level sets. The algorithm
partitions a large cuboid containing the knot into small cubes; then, on each
cube, it uses a bilinear approximation of that component. If the bilinear
approximations of all three components of $\pmb{E}$ pass through zero in the
same cube, then we take the center of that cube as an initial guess for our
multivariable Newton method. Some initial guesses to Newton end up diverging
or converging to a far away critical point, and we throw these away. In
contrast, the successful initial guesses quickly converge to the approximate
locations of the zeros of $\pmb{E}(\pmb{x})$.

Along with the zeros, we are also interested in the equipotential surfaces.
These are given by $\phi^{-1}(v)$, where $0<v<\infty$ is some given voltage
level. The relevant values of $v$ range from small positive values far from
the knot, to large positive values close to the knot. Let $v^{*}=\phi(x^{*})$
denote the potential at an equilibrium point. Recall that equipotential
surfaces undergo self-collisions at $x^{*}$ and lose smoothness there. So to
get a smooth surface, we perturb $v^{*}$ to a nearby regular value $v$ at
which the Hessian matrix of second derivatives of $\phi$ (equivalent to the
Jacobian of $\pmb{E}$) has full rank. By the implicit function theorem, this
full rank condition ensures that $\phi^{-1}(v)$ is a smooth, orientable,
compact surface without boundary. We then use the Marching Cubes algorithm to
render the surface. By repeating this process for a range of $v$ values, we
can explore how the equipotential surfaces change as we vary the level of the
potential.

###  III.4  Results and Conjectures for Charged Knots

To illustrate the results obtained with this approach, consider the following
parametrization of a trefoil knot:

$\pmb{r}(t)=(\sin{t}+2\sin{2t},\cos{t}-2\cos{2t},-\sin{3t}).$ (4)

In our numerical simulations, we sampled a cubic domain of $30\times 30\times
30$ initial guesses in a mesh surrounding the knot and ran the multivariable
Newton method to test for convergence to a zero. We rejected iterations that
grew too large, or were within a small distance threshold from another
computed zero, indicating a duplicate.

Figure 10 shows that the electric field has seven zeros for this particular
embedding of a trefoil. By calculating the eigenvalues at these zeros, we can
confirm that they are all saddle points and classify them by their indices
(the dimensions of their stable manifolds).

Then, to obtain representatives of the equipotential surfaces around the
trefoil, we compute the critical values $v^{*}$ at the zeros, perturb them to
nearby values $v$, and take their inverse images $\phi^{-1}(v)$. For the
parametrization (4), we find the outer triplet of zeros in Figure 10 has
$v^{*}\approx 12.79$ and indices of 1; the inner triplet has $v^{*}\approx
15.82$ and indices of $2$; and the origin has $v^{*}\approx 15.42$ and index
$1$. To see what the equipotential surfaces look like in between these
critical cases, we compute the level sets $\phi^{-1}(v)$ for the perturbed
values $v=12.7,15,15.5,$ and $16$. Figure 8 shows the resulting surfaces.
Topologically, they are knotted tori with various numbers of holes.

Figure 10: A top-down view of the locations of the zeros (red dots) in the
electric field around a uniformly charged trefoil knot.  Figure 11: The number
of zeros in the electric field around a figure-eight knot as it is flattened.
Here, the knot is parametrized as $x(t)=(2+\cos 2t)\cos 3t$, $y(t)=(2+\cos
2t)\sin 3t$, $z(t)=\gamma\sin 4t$ and we vary the height $\gamma$.

In the example above, we assumed a highly symmetrical parametrization of a
trefoil. What happens if we break the symmetry or, more generally, if we
deform a knot continuously without allowing it to pass through itself? How
does that affect the number of zeros in the electric field around the knot?
Ideally, we would like to find deformations that cause as many zeros as
possible to coalesce, thus bringing us closer to the absolute minimum number
of zeros, whatever that may be, and perhaps allowing us to improve on the
tunnel number bound.

One strategy is to deform the knot so as to reduce its complexity in some way,
as quantified by an energy functional or a more general Lyapunov function. Two
energy functionals in the literature on physical knot theory are the Möbius
energy [49] and the Buck–Orloff energy [50], whose locally minimal
configurations enjoy nice regularization properties. But we have found a
simpler strategy to be useful: we slowly squash the knot from the top down and
watch what happens to its zeros.

Figure 11 shows that by flattening a figure-eight knot we can reduce its
number of zeros from nineteen to five; then, unexpectedly, the number goes
back up to nine. It is intriguing that the number of zeros can either increase
or decrease as one flattens the knot. The staircase structure of the graph
reveals that zeros can appear or disappear in pairs (which is what one expects
generically [6]) or in two simultaneous pairs (due to non-generic symmetries
in the particular parametrization of the knot). We suspect that five is the
smallest number of zeros possible for _any_ parametrization of a figure-eight
knot, symmetrical or otherwise.

Let $Z$ denote the minimum number of zeros around a charged knot $K$, where
the minimum is taken over all smooth embeddings of $K$. Although we have not
found a formula for $Z(K)$, we conjecture that it can be bounded from above
and below by two standard topological invariants of $K$:

$2t+1\leq Z\leq 2c+1.$ (5)

We have already met the $2t+1$ lower bound, which uses the knot’s tunnel
number $t$. This lower bound has been proven [7]. By contrast, the $2c+1$
upper bound is conjectural. It involves the knot’s crossing number $c$,
defined as the minimal number of crossings possible in a planar projection of
the knot. We have numerical evidence and a plausibility argument but not a
proof, for reasons we will explain momentarily. Table I lists these bounds
along with our conjectured values of $Z$ for some simple knots.

Knot | $2t+1$ | Conjectured $Z$ | $2c+1$
---|---|---|---
Unknot | 1 | 1 | 1
Trefoil | 3 | 7 | 7
Figure-eight | 3 | 5 | 9
Cinquefoil | 3 | 11 | 11
Three-twist | 3 | 11 | 11
Table 1: Conjectured minimum number of zeros $Z$ for knots with five or fewer
crossings. In each case, our conjectured value of $Z$ lies between the proven
lower bound $2t+1$ and the conjectured upper bound $2c+1$. Our conjectures are
based on numerical experiments that used the following knot parametrizations:
(1) Unknot: $x(t)=\cos t$, $y(t)=\sin t$, $z(t)=0$. Trefoil: $x(t)=\sin
t+2\sin 2t$, $y(t)=\cos t-2\sin 2t$, $z(t)=-\gamma\sin 3t$. Figure-eight:
$x(t)=(2+\cos 2t)(\cos 3t)$, $y(t)=(2+\cos 2t)(\sin 3t)$, $z(t)=\gamma\sin
4t$. Cinquefoil: $x(t)=(\cos 2t)(3+\cos 5t)/2$, $y(t)=(\sin 2t)(2+\cos 5t)/2$,
$z(t)=(\gamma\sin 5t)/2$. Three-twist: $x(t)=2\cos(2t+0.2)$,
$y(t)=2\cos(3t+0.7)$, $z(t)=\gamma\cos 7t$. The standard parametrization has
height $\gamma=1$. To find a plausible conjecture for the minimum number of
zeros, we flatten the knot by slowly decreasing $\gamma$ to $0$, and watch the
zeros coalesce.

Our plausibility argument for $2c+1$ as an upper bound on $Z$ is based on the
following observation: For any knot $K$, we can always construct an embedding
of $K$ that has _at least_ $2c+1$ zeros in its electric field. Unfortunately,
that statement is weaker than what we need. To prove that $2c+1$ is an upper
bound on $Z$, we would need to construct an embedding with _exactly_ $2c+1$
zeros. We suspect that this stronger construction is always possible too, but
we have not proven that yet. For now, we outline the main ideas behind our
weaker construction [6], which proceeds in three steps.

First, we flatten $K$ into a plane curve. The resulting curve crosses itself
and bounds a number of planar regions that we call holes (more properly,
“holes” are bounded connected components in the planar complement of the
flattened knot). The electric field in $\mathbb{R}^{3}$ produced by a charged
planar curve has a special property: at all points in the plane away from the
curve, the electric field vector $\pmb{E}$ lies within the same plane.
Moreover, at points very close to the charged curve, the direction of
$\pmb{E}$ is nearly perpendicular to the curve (because it receives its
dominant contribution from the portion of the charged curve nearby). Thus the
winding number of the electric field around the boundary of each hole is $1$.
The Poincaré–Hopf index theorem then implies that each hole contains _at
least_ one source or sink zero of the planar vector field. These planar zeros
extend to zeros of the full electric field in $\mathbb{R}^{3}$, because the
out-of-plane component of $\pmb{E}$ is also zero, as discussed above. In Ref.
[6] it is proven that these zeros are all saddle points of index $1$ in
$\mathbb{R}^{3}$. A counting argument then shows that a planar curve with $c$
crossings has $c+1$ holes, and since we just showed that each hole must
contain at least one zero, we arrive at our first conclusion: the electric
field around a flattened knot has at least $c+1$ zeros.

But we are not done yet. A flattened knot is not an admissible knot because it
has self-crossings. So the second step is to perturb the flattened knot by
lifting one of its strands up out of the plane, ever so slightly, at each
crossing to restore the topology of the original knot. By performing these
lifting operations in tiny neighborhoods of the crossings that are
sufficiently far away from the aforementioned zeros, we are sure to preserve
the existence and topological types of the $c+1$ (or more) zeros deduced in
the first step, thanks to the structural stability of gradient vector fields
[51].

Now comes the third and final step. By applying the Morse inequalities [52],
one can show [6] that each lifting performed in the second step gives rise to
a new zero at the associated crossing, and this zero is of index 2. Since a
total of $c$ liftings are performed, altogether we get $c$ additional zeros of
index $2$. Counting the $c+1$ (or more) zeros of index $1$ and the $c$ zeros
of index $2$, we conclude the electric field around a squashed but not
strictly planar version of $K$ has at least $2c+1$ zeros.

The weakness of the “at least” part of the conclusion can be traced back to
the Poincaré–Hopf index theorem; that is where the first “at least” qualifier
popped up. If we could ensure _exactly_ one zero in each hole, we would then
be able to claim what we want: $Z\leq 2c+1$. We suspect the uniqueness of the
zero in each hole would follow if the holes were round enough (neither too
elongated nor too non-convex), but this is what remains to be properly
formulated and proven in future work.

##  IV Discussion

As we have tried to show in this article, the study of knotted charge
distributions opens up many new directions for exploration. The questions are
mainly motivated by their conceptual simplicity and theoretical appeal, but
they could have real-world implications. For instance, given that the zeros of
electric and magnetic fields are relevant to problems of plasma confinement in
nuclear fusion and to trapping of cold atoms, related questions may be of
experimental interest in those settings [53, 54, 55]. Charged knots also arise
in molecular biology and polymer physics, where researchers study the
knottedness of charged DNA molecules and their interactions with electric
fields [2, 3, 4, 5].

###### Acknowledgements.

We thank Greg Buck for helpful discussions. M.L. was supported in part by an
NSF RTG Grant DMS-1645643. A.T. was supported by National Science Foundation
grants DMS-1818757, DMS-1952757, and DMS-2045646.

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