Saturday, 16 September 2017

Extreme High-Speed Dreidel Physics

(Warning: if you're viewing this post on a phone with limited should probably leave before all the gifs load)

Every November or December, Jews around the world celebrate Hanukkah, a holiday that commemorates defeating and driving out the Hellenistic Seleucids from Israel and establishing the short-lived Hasmonean kingdom* around the year 160 BC. The celebration of Hanukkah involves lighting candles, eating greasy food, exchanging gifts, re-telling the story, and playing games with a spinning dreidel.

Legend has it that when the study of Judaism was banned by the Seleucids, scholars took to the hills to study in secret, and when authorities came to investigate, they pretended they were just doing some good wholesome gambling. That legend, however, is very likely made up and doesn't appear in print before 1890. I read an article arguing that the dreidel originated from a popular gambling device called a teetotum that was popular in Northern Europe in the medieval period, that eventually made its way into Yiddish culture. The letters on the four sides of the dreidel (equivalent to N, G, H Sh) are interpreted to stand for the Hebrew "Nes Gadol Haya Sham" (a great miracle happened there), but originally stood for the Yiddish rules for gambling with a dreidel: nit (nothing), gants (everything) halb (half), and shtel ein (put one in)**.

From a physics perspective, a dreidel is an example of a spinning top, a source of extremely difficult homework problems in undergraduate classical mechanics related to torque and angular momentum and rigid body motion and whatnot. I was chatting with a theorist I know who mentioned that it would be fun to calculate some of these spinning-top phenomena for the dreidel's specific geometry (essentially a square prism with a hyperboloid or paraboloid base), and I suggested trying to compare it to high-speed footage. A quick literature review revealed that most of the dreidel analysis has to do with whether the gambling game is fair and how long the games last. Annoyingly, the search was obfuscated by the fact that there's a publisher called DReidel.

My lab has a high-speed camera that is used to film gel particles and droplets as they deform. It is normally connected to a microscope, but with the help of a student in my lab, we removed it and connected it to a macroscopic lens we had lying around in ye-olde-drawer-of-optics-crappe. A representative of MIT Hillel graciously provided me with a few dreidels, and I some time spinning the dreidels in front of the high-speed camera and recording them at 1000 frames per second.

Before I get into the more quantitative analysis, let's just take a look at what a dreidel looks like in slow motion, because as far as I can tell from a the google, I am the first person to attempt this.

As I initially spin the dreidel, it spins in the air a few times, lands with an axial tilt, and gradually rights itself as its angle of precession comes closer to the vertical. After that, you can see it spinning rapidly and precessing a little bit, but not doing anything too crazy.

The self-righting behaviour is a lot more extreme when I do the advanced-level upside-down flip.

On those first few bounces, it really looks like it's going to fly out of control and crash, but still it gradually rights itself into a stable rotation. While this self-righting tendency is is not unstoppable.

It's also pretty fun to watch what happens when they eventually lose stability and fall over.

This self-righting is too complicated-looking for me to understand right now, so let's look at something simpler: the steady-state (a) rotation and (b) precession.

It rotates several times while precessing from left to right.

To perform quantitative analysis of the dreidel's motion, I would want to be able the measure the phase of its rotation over time. Because the dreidel is made of reflective plastic, as it spins it reflects light into the camera, moreso when its face is parallel to the plane of the camera. Thus by measuring the total intensity, we should have a proxy for the phase of the dreidel, each intensity peak being a quarter-turn, and can investigate how that evolves over time. I wrote a MATLAB script that summed the total intensity of each frame and plot it over time.

There was initially a problem with this method of analysis, however. You can figure it out if look at the wall behind the dreidel in the above gifs (especially the wide crashing one), and notice that it's flickering. This is because the room is illuminated with AC electric light with a 60 Hz frequency. The light intensity is proportional to the square of the current, so it has a maximum twice per cycle, and the light flickers at 120 Hz. That is exactly the frequency at which the intensity fluctuates; the flickering was swamping the contribution from the dreidel. However, the quarter-turn frequency isn't that far off, so I was getting some neat beat frequency dynamics as well***.

This caught me off guard and it was skewing all my videos, so I took another few recordings using my cell phone flashlight with all the AC lights turned off. The videos don't look nearly as good, but the time-series are cleaner.

We can the intensity fluctuating periodically every ~8 ms, corresponding to a rotation period of 32 ms (nicely close to the square root of 1000, so it's also around 32 rotations per second), and a slower mode of about 200 ms or 5 precessions per second. 32 rotations per seconds is 128 quarter-rotations per second, so you can figure out why it took me a while to figure out that I had to disentangle it from the 120 Hz light flickering.

The fourier transform shows two peaks, one corresponding to rotation and one to precession, which is of stronger amplitude (I believe this is due to my analysis method and not to actual physics). The reason the peaks are smeared and not as at a sharp frequency is because the angular velocity gradually decreases as the dreidel loses energy to friction, so the peaks get smeared to the right.

With the flickering out of the way, I can also calculate how the rotation period evolves over time, using a peak-finding function in MATLAB. It gradually gets slower, as expected, which 6-7 ms between peaks at the beginning, and 14 ms between the peaks before it crashes. If this is caused by dry friction at the base, we would expect the frequency to decrease linearly with time. If it's caused by viscous drag, we would expect an exponential decrease. What do we see? The fact that it's discretized by the frame rate makes it harder to tell, but applying a rolling average on the frequency decrease suggests that it is linear and thus caused by dry friction.

The measurement gets a bit hairy towards the end.

That was mostly a discussion of the rotation, although precession presented itself as well. Let's take a closer look the precession. I wanted to measure the angle the dreidel was at with respect to the vertical, and how that evolved over time. This is not as easy to measure as the total image intensity; I had to use Principal Components Analysis. I found an algorithm on this blog post, and it worked as follows:

  1. Define a threshold intensity such that everything in the image above the threshold is dreidel, and everything below it is background. Set the dreidel equal to 1 and the background equal to 0.
  2. Define two arrays, one containing all the x-coordinates of all the 1's, and the other containing all the y-coordinates of all the 1's (such that each 1-pixel is represented in the array).
  3. Calculate the 2x2 covariance matrix of the location array.
  4. Find the eigenvectors and eigenvalues of the covariance matrix.
  5. Find the angle associated with the eigenvector with the larger eigenvalue (e.g. the arctangent of the ratio of its components)
With some finagling to get the angle in the correct quadrant (I wanted the angle from the vertical, not from the horizontal), this works pretty robustly. It works better if I crop the movie to only include the stem. This actually only measures the angle of the dreidel projected onto the plane of the camera and doesn't provide information about its tilt towards or away from the camera, but it's good enough to see how the precession angle evolves. I can examine how the angle evolves over time...and it's pretty neat.

Two things are apparent from looking at this graph: both the amplitude and frequency of precession are increasing over time. The fourier spectrum of the precession angle contains only the precession peak, without the rotation peak at higher frequency. What's happening is that gravity is exerting a torque on the dreidel at an angle relative to its principal angular momentum vector, which induces a precession in the direction determined by the cross product of spin and down. The angular frequency of precession is inversely proportional to that of rotation, so that as the dreidel slows due to friction, its precession speeds up, which is what we see. The spinning is essentially preventing the gravitational torque from pulling the dreidel down, and as it loses angular velocity, the precession angle gradually increases.

This whole project started as a discussion with a colleague about how the term "Jewish physics" should be repurposed from a label the Nazis used to discredit Einstein, and dreidels seemed like a natural thing to focus on. After fiddling around with a high-speed camera for a bit I got some cool videos, and thought I'd share them. I didn't really cover anything in this post that isn't explained in great detail in dozens of analytical mechanics textbooks, but it's perhaps the first time anyone has used a high-speed camera to record a dreidel. I thought it was neat.

In addition to being a fun little diversion, it also spurned an improvement in my DNA image analysis code. It was taking so long to open the dreidel movies in MATLAB that I looked into a more efficient way of doing so, which improved loading time by like a factor of 20 (from 55 seconds down to 3 seconds), which I now use to open DNA movies as well.

*If you grew up hearing the Hannukah story every year, you probably will not recognize the words Hellenistic, Seleucid, or Hasmonean.

**For those unfamiliar with Jewish linguistics, Hebrew is a Semitic language related to Arabic and Ethiopian (Amharic), whereas Yiddish is a Germanic language that uses Hebrew letters, so some of the words are similar to the also-Germanic English, e.g. halb and half.

***This is what the FAKE NEWS wrong analysis looks like:

Thursday, 23 February 2017

Comfortable Relativistic Space Travel: New paper on arXiv

I have written a new paper which has just come out on It has the somewhat long title of  "Relativistic minimization with constraints: A smooth trip to Alpha Centauri" and can be found here. My co-author on this paper was Riccardo Antonelli, who also writes a physics blog, called Hologrammata. It discusses and derives the most comfortable way to plan a relativistic journey, which may become relevant in several thousand years when we can actually go fast enough for this to matter.  In this post, I will explain what this paper is about and what inspired me to write it.

Artist's impression.

Every month, the American Journal of Physics puts out a new edition with about twenty articles. These are usually a delight to read and can be read by anyone with a basic physics background. Some of the articles are about physics education, but most of the research articles involve applications of established physics to problems nobody had quite thought of before. (This is where my tunnel-through-the-Earth paper is published).

In the October 2016 edition, I read an article called "The Least Uncomfortable Journey from A to B" (possible free version here) by three Swedish and possibly French researchers, Anderson, Desaix, and Nyqvist. In it, they ask the following question: on a linear trip taking time T from point A to point B separated by distance D, what is the trajectory (instantaneous velocity as a function of time or distance) that minimizes the  time-integrated squared acceleration? If you can't mentally convert that sentence into LaTeX, the "discomfort" here is the total acceleration that the passenger feels throughout the trip, pushing them into their seat on the first half of the trip, and thrusting them out of their seat on the second half.

The solution, which they find through Lagrangian-style variational calculus, turns out to be a trajectory with constant jerk, where the acceleration decreases linearly over the entire trip, such that the velocity is a quadratic function of time. A lot of the paper talks about various analytical approximation methods, and also computes an alternate "least uncomfortable" trajectory where the squared jerk is minimized instead of the squared acceleration.

The interesting thing about the solution is that there is only one. The relative velocity (v/vmax) as a function of the relative position (x/D) or time (t/T) does not actually depend on the distance or time. No matter how far or fast you want to go, there is only one set least-uncomfortable way to get there. However, there is a problem with this. If you want to calculate how to plan a train ride that covers 500 miles in an hour, the solution is fine. But if you want to plan a trip that goes from Earth to Jupiter in an hour, then it starts running into some problems. Although it wasn't stated in the paper, the solution only holds when D/T is  much less than the speed of light. I was curious about finding a relativistic solution to this problem, and if I get curious enough about a problem that is actually possible to solve, I can be quite persistent.

In the relativistic version of this problem, when you consider the integral over time of the squared acceleration, it matters in which reference frame the time and acceleration are being measured. Because we care about the comfort of travellers and not of stationary observers, we consider the proper acceleration and not the coordinate acceleration. This problem can be formulated either in terms of "lab" time or proper time on the ship, there are benefits to both. My first attempt at this essentially involved following Anderson et al.'s derivation except with a lot of extra Lorentz factors, which is generally not a good way to go about things, and I stalled pretty quickly.

Then one day, I was talking to my internet friend Riccardo, who had just finished his master's thesis in string theory and holography. I figured a lot of the relativistic analytical mechanics formalisms might still be fresh in his mind, so I mentioned the problem to him. He too found the problem interesting, and came to the realization that if the problem was formulated in terms of the rapidity (the hyperbolic tangent of velocity relative to c) since its derivative is proper acceleration, it could be expressed as a much neater problem than my "add a bunch of Lorentz factors" approach.

The way to derive the solution, Riccardo discovered, is to treat it as a Lagrangian function of the rapidity rather than the position (such that you write down L(r, $\dot{r}$, ) instead of L(x, $\dot{x}$) and apply the Euler-Lagrange to higher-order derivatives of position than normal). Even though we were unable to derive a closed-form solution, it turns out, the rapidity of the least uncomfortable solution evolves like a particle in a hyperbolic sine potential, and I was able to generate solutions using the Runge-Kutta algorithm, which has been serving me well since I learned how to use it in 2008.

As I said, there is only one classical solution, it is universal for all distances and times. However, when relativity it taken into account, the solution depends on how close to the speed of light it gets: there is now a single free parameter that characterizes the solution, and we can express it in terms of its maximum velocity, rapidity, or Lorentz factor. Our solution recovers the classical solution in the low-velocity limit (which is good, otherwise we'd be obviously wrong), but as the maximum speed increases, a greater fraction of the time is spent close to the maximum (which makes sense, as the path is some form of "accelerate close to light speed, decelerate down from close to light speed" and as you approach light speed, your velocity changes less and less)

A figure from the paper showing relative velocity as a function of relative time. The classical solution is the inner parabola with the car next to it, the solutions get faster as you go out, the fastest has a maximum Lorentz factor of 100 and is marked by a rocket. While making this figure I had a minor legal quandary about the copyright status of unicode emojis.

Now, we had solved the problem that I was inspired to solve, the relativistic version of Anderson et al.'s least uncomfortable journey. However, this whole thing may be moot for space travel: you don't necessarily want to keep acceleration to a minimum, instead you might want to contintually accelerate with Earth-like gravity for the first half of the trip, then reverse thrust and decelerate with Earth-like gravity for the second half, so that the entire ride (besides the switcheroo in the middle) feels like Earth. We calculated a trip like this to Alpha Centauri, which would take six years relative to Earth and 3.6 proper years on-board due to time dilation, reaching 95% light speed. With our solution, covering that distance in the same proper time would only reach 90% the speed of light, and might be more appropriate for sending mechanically sensitive electronic probes or self-replicating machines than it would be for sending people.

Anyway, we wrote up the paper and now it's online for the world's reading pleasure. It was fun seeing this paper go from a persistent idea to a finished product in a few weeks time, and dusting off some of the long-forgotten special relativity tools. It was a nice little collaboration with Riccardo, where I came up with the problem and he came up with the solution. This is the first random physics idea I've had in while that has actually come to fruition, and I hope there are many more to come.

Friday, 3 February 2017

Saturday, 14 January 2017

Empirical Searches for Nominative Determinism

Nominative determinism is the idea that a person's name influences their life or career. The Seven Dwarfs are an example, and humorous lists have been written about people with names that suit their vocation. The world's fastest man is named Bolt, there was a Russian hurdler named Stepanova, a meteorologist named Julie Freeze, and eugenics pioneer Eugen Fischer. Considerable attention is given to a urology paper by Weedon and Splatt.

To my knowledge, the idea is not taken too seriously. However, I was recently curious about the extent to which people have looked for evidence of nominative determinism. I went on Google Scholar and found some results which I'm sharing with you now.

Most analyses of nominative determinism focus on medical professions, looking at whether specialists tend to have names associated with their speciality.

Examples of nominative determinism in medical research, with statistics of N=1. Ref.
Although this study was not entirely serious, it had a number of methodological flaws as it failed to consider the prominence of each surname in the medical fields relative to its prominence in the population at large.

One recent study was by four members of a family surnamed Limb, all doctors (two of whom named C. Limb are avid climbers). They surveyed a directory of over 300,000 medical doctors grouped by speciality, and went through the list and independently noted which were suited to medicine in general and to their speciality. The lists were then merged by consensus. They found that 1 in 149 doctors had medically appropriate surnames, and 1 in 486 had names relevant to their field. They noted:

"Specialties that had the largest proportion of names specifically relevant to that specialty were those in which the English language has provided a wide range of alternative terms for the same anatomical parts (or functions thereof). Specifically, these were genitourinary medicine (eg Hardwick, Kinghorn, Woodcock, Bell) and urology (eg Burns, Cox, Dick, Koch, Cox, Balluch, Ball, Waterfall). Some terms for bodily functions have been included in the frequency counts but are not mentioned because they are colloquial terms that may lower the tone of this publication."
Another study looked at nominative determinism in patients rather than doctors, trying to see if surname influences health. Based in Ireland, the authors looked through the Dublin phone book to estimate the percentage of people surnamed Brady (0.36%), and then tried to ascertain what fraction of them had been diagnosed with bradychardia (slow heart rate). The only data they had available was for pacemaker implants, and found that 8 out of 999 pacemakers had gone to people  named Brady. Despite the small numbers, this was statistically significant (p=0.03) relative to Brady's prominence in the Dublin telephone directory. The authors don't speculate on the causes of this, but neglected to discuss the idea that doctors may be subtly biased towards diagnosing Bradys with bradycardia.

"This increased bradyphenomenon in Bradys could be attributable to increased levels of bradykinin"

In the analysis of particle physics experiments there is something called the "look elsewhere effect" where any significant effect has to be considered in light of all the other effects that are being looked for (this xkcd gives an example of the folly), which reduces the overall significance of the interesting part. I suspect that if they had looked at what other surnames were over-represented in bradycardia diagnosis, the significance of Brady would be diminished. I think this proviso should be applied to any search for nominative determinism: look at the boring names as well as the silly ones.

There are a few other studies out there, mostly tongue-in-cheek. These do go a step beyond compiling humourous lists, but not too much further.

Sunday, 4 December 2016

Strange Moments in Science: The Battle for the Planck Length

Many people use Wikipedia as a go-to source of information on a topic. People know that anyone can edit Wikipedia at any time, but generally trust that it is largely accurate and trustworthy because of the large number of vigilant volunteers and the policy of requiring sources for facts. A few years ago, I noticed that the article on the Planck length had some very detailed yet weird and incorrect information that was causing a lot of confusion among physics enthusiasts, and I decided to look into it.

This image taken from the Wikipedia commons and based on an xkcd.

I have written about the Planck length and its physical meaning on PhysicsForums. Besides being a "natural" unit, I stressed that the main significance of the Planck length is that it is approximately the order of magnitude at which quantum gravity becomes relevant, but that the idea that it makes up some sort of "space pixel" is a common misconception. There may be additional significance to it, but that wouldn't be part of established physics.

The Wikipedia page on the Planck length made the claim that a photon with a Planck length wavelength would collapse into a black hole. This is physically problematic (what does it even mean for a photon to collapse into a black hole?) in part because it violates Lorentz symmetry, because you could just observe the photon in a different reference frame and it would have a longer wavelength. Would the black hole "uncollapse" if you moved away from it? Is it only a black hole in some privileged reference frame. There was a large "proof" of this purported fact based on the Schwarzschild metric and the gravitational self-energy of a photon (which doesn't really make sense either because photons don't have a rest frame...nor are they uniform spheres). If you're using general relativity to contradict special relativity, you have done something wrong.

I thought this was strange, so I wanted to look at the source, which was a paper with a Russian title that in English meant "Comprehending the Universe." I did a search for it, and all that came up was the same Wikipedia page. The author of the referenced paper suspiciously had the same name as the person who had made the majority of the recent edits to the Wikipedia page. The reference is published by Lambert Academic Publishing, which is widely known to be a scammy vanity press that will publish your manuscript and then sell you a paper copy. They have recently been emailing me asking if I want them to publish my thesis. I was suspicious and perturbed, so I made a comment on the Wikipedia talk page mentioning that the reference couldn't be found. The author then posted a link to a page where it could be purchased for $60.

So basically, the situation was that there was this incorrect information being passed off on Wikipedia as "proven" physical truth, based on an article published in Russian in a vanity press and added to Wikipedia by the author. This was Bad, and it was misleading people, and I wanted to fix it. It can be quite difficult to actually change things on Wikipedia, because of the large number of policies that have to be followed and protectively zealous editors who will revert a lot of the changes they see. I figured if I just started removing the relevant material, it would quickly be undone by this guy who wanted his stuff on Wikipedia. So, I went to a friend who I knew to be a pretty serious Wikipedian, who goes by the nickname Oreo Priest.  I explained the situation to him, and he logged in to back me up. He pointed out on the talk page the various policies that were being violated (original research, reliable source, conflict of interest, etc) and gave me carte blanche to remove the offending material.

So, I went ahead and started removing it, and when the author protested, my friend and some other Wikipedians backed me up by citing official policies. The author's final argument was that his work was presented at the "5th International Conference on Gravitation and Astrophysics of Asian-Pacific Countries" in Moscow in 2001, and was therefore legit. Then, the final smackdown was laid:

The fact that you also presented your work at a conference doesn't change any of the above points, sorry. 
After that, the pollution stopped, although I haven't checked the guy's profile lately to see what he's up to. A large chunk of the article was removed, and hasn't been added back. However, there hasn't been a lot of improvement to the article since, so the "Theoretical Significance" section is a bit disjointed and sparse. It, and some of the other Planck unit articles, could use the attention of some quantum gravity researchers. Still, it's much better than the page on topological insulators. That page is terrible.

Tuesday, 29 November 2016

The Bremen Physics Catapult

In the town of Bremen in Northwest Germany there is a facility where researchers can launch their experimental apparatus by catapult. It is at a research facility called ZARM, the Center for Applied Space Technology and Microgravity (the acronym makes more sense in German).

The tower in all its glory (from the ZARM website).

The main goal of this facility is to provide microgravity environments for physics experiments, on a scale more accessible than orbital space launch or parabolic aircraft. Besides answering the kinds of questions that keep us up at night, like how cats behave in zero gravity, microgravity is a useful method to control the effective gravitational field in an experiment. As Albert Einstein reminds us, true freefall is indistinguishable from inertial rest, and remaining static in a gravitational field is indistinguishable from uniform acceleration. If you want your experiment to take place in a truly inertial reference frame, dropping it down a tower is a simple way to achieve this.

The entire experiment is built into a capsule like this, and then launched.

Initially, it was simply a drop facility: an experiment built into a modular capsule could be raised by an elevator and dropped from the top of the tower, falling 120 meters and experiencing four seconds of freefall. In 2004 the facility was modified to include a catapult launch with an initial speed of 168 km/h, giving it twice the freefall time but making the facility at least eight times as cool.

A gif of a lanch and landing, pillaged from this youtube video.

Based on a few videos of the facility, it appears that immediately after launch, a crash pad full of protective foam slides into place, protecting the experiment as it lands. Some of the videos are quite well done, check them out if you're interested.

Launch and landing.

I first became aware of this experiment when I read a paper (or possibly a news article) about launching a Bose-Einstein condensate in the catapult. In quantum mechanics class, we learn that a system whose wavefunction that is initially a Dirac delta will expand isotropically through time-evolution, becoming more spread out. Bose-Einstein condensates, functionally made of cold atoms in an electromagnetic trap, serve as a good experimental system for wavefunction dynamics because they are much larger than an individual particle while still behaving like one. However, none of those quantum mechanics homework problems included an anisotropic acceleration vector like we have on Earth's surface, and breaking the symmetry like that causes the condensate to expand differently than it would in true freefall. To measure this predicted isotropic expansion, the research group launched their experimental from a catapult. Because these Bose-Einstein condensates are so sensitive to acceleration, the group was proposing they could be used for accelerometers in spacecraft navigation systems. We will see if that actually happens.

The Bose-Einstein Projectile, from the Humboldt University group.

Not every experiment is as exotic, sometimes it's used to characterize apparati that will eventually be launched on satellites. Some experiments are testing the expansion of explosions in microgravity rather than atomic condensates, both for science's sake and also to gauge safety protocols aboard spacecraft. There are also some biological experiments, for example studying how fish orient themselves and turn in microgravity (I guess they couldn't get the approval to use cats), and others to test how plants perceive gravity (I feel like "how to plants know to grow up" can become a pretty complicated question. The projects are listed on the ZARM website, some in English and some in German.

My experiments typically take place in a little pocket of fluid on a piece of glass on top of a microscope on an optical table in a stationary lab. Compared to launching an experiment from a catapult, that seems rather mundane.

Thursday, 3 November 2016

DNA, topological ropelength, and minimal lattice knots

This post is about some diversions into knot theory, a branch of topology, that I took while contemplating my experiments. My job involves performing and analyzing experiments where I stretch out DNA molecules with knots in them. We study DNA because it serves as a good model polymer that's analogous to the much smaller polymer molecules that plastics and other modern materials are made of, and we study knots because we want to understand how entanglement affects polymer dynamics. There are also some genetic sequencing technologies for which we might want to add or remove knots, but right now I'm mainly just trying to figure out how polymer knots work. The theoretical side of the project (largely carried out by my colleague Vivek and Liang) involves running simulations of polymer chains with different kinds of knots, or simulating the formation of knots, and thinking about that is where I start to dip into knot theory.

A DNA molecule with two knots in it, from one of my experiments. The molecule is about 50 microns when stretched.

Mathematically, knots are only truly defined in loops, as a knot in an open string can just be slid off the end. Knots are loosely categorized by the minimum number of times the loop crosses over itself in a diagram (see below). The simplest knot, the trefoil, when drawn must cross itself a minimum of three times. For a given crossing number, there can be multiple kinds of knots. The two simplest knots, with three and four crossings, only have one type, then there are two for five, three for six, seven for seven, then it starts to explode:  165 for ten crossings, 9988 for thirteen, and over a million for sixteen. Beyond this, it is unknown how many kinds of knots there are (although there are constraints). Each type of knot has certain parameters that are unique to the  knot, called invariants, and calculating these invariants is how knots can be distinguished. When dealing with knots in real ropes and polymers and DNA, we have to remember that it doesn't count as a knot unless the ends are closed. If the ends are very far from the interesting knotty part we can just pretend this is the case without too much issue; in simulations of tightly bunched knots however there are various algorithms called "minimally interfering closure" that make knots in open strings topologically kosher.

Knots with up to seven crossings.
The typical model of a polymer is that of a self-avoiding walk, which I've talked about in the past. When the walk is forced to have closed ends, it represents a topologically circular molecule, and can be thought of as a self avoiding polygon (even though polygons are typically imagined in two dimensions). These self-avoiding polygons can take the form of different types of knots, and in fact it can be shown that as a self-avoiding walk becomes longer and longer, the probability of having a knot in it approaches 100%. A number of computational papers have shown that knotted ring polymers have a size (quantified by the radius of gyration, the standard deviation of the location of each monomer in the chain) which is smaller than an unknotted chain of the same length.  This makes intuitive sense, there are topological constraints which prevent the chain from adopting a widespread conformation.

The chains in these simulations tend to be quite "loose," and I was curious about the scaling of tight or  "ideal" knots, which are knots that have the minimum possible length-to-diameter ratio. Each type of knot has a parameter called the ropelength, which is the minimum length required to make a knot of unit diameter. The ropelength can be a difficult quantity to calculate, and one group has written a few papers just slightly hammering down the ropelength of the simplest knot, bringing it from 16.38 in 2001 to 16.37 in 2014.

These knots are "ideal" meaning they have the smallest possible length-diameter ratio.

The ropelength comes up in my research in part because of a theory from Grosberg and Rabin, who derived the free energy of a knot in a polymer. The free energy arises from the fact that the length of polymer in the knot is confined, reducing entropy, and bent, increasing energy. They argued that the energy of a given knot type is minimized at a certain length, and such a knotted conformation constitutes a metastable local minimum in the free energy (compared to the unknotted global minimum) that implies that an open knotted polymer wouldn't spontaneously unknot itself, but instead would unknot through a slower diffusion process when the knot reaches one end. One of the terms in the Grosberg-Rabin is the excess knot contour, which is the length of polymer in the knot minus the ropelength, and thus it is necessary to know the ropelength to calculate the metastable knot energy. Another paper argued that metastable knots don't exist, and that is one of the things I'm trying to figure out with my knotted DNA experiments.

The most topologically accurate DNA knot experiment comes from Robert Bao, Heun Jin Lee, and Stephen Quake in 2003, where knots of known topology were tied in individual DNA molecules using optical tweezers. I get the impression that Bao developed the expertise to do this, wrote his paper, and made the sanity-promoting decision never to do it again. In their paper, they measured the amount of DNA in different kinds of knots and the diffusion of the knots along the molecule, and used this to ascertain how much intramolecular friction was occurring within the knot.  Grosberg and Rabin claimed to have postdicted Bao et al.'s findings with their model. In my experiments, unfortunately, the knots are much more complicated and I don't know what kind they are.

In my experiments, I'm interested in figuring out what kind of knots I'm looking at, and one way to do that is to estimate the amount of DNA in the knot and backstrapolating to figure out the number of essential crossings. To aid in this endeavor, I looked up the known ropelength of knots up to 11 crossings, which was not super-easy nor super-difficult to find. I don't think anyone has analyzed this specific relationship before, but not too surprisingly, with more crossings you tend to have a longer knot, although there is overlap between knots with adjacent or superadjacent crossing number. This relationship fits well with either a linear or square-root function (the best-fit power is 0.8...not sure that's meaningful but I love power-law fits). Once you get past 11 crossings there are problems because A. nobody has bothered calculating this for 12-crossing knots (of which there are 2176) and B. the mean isn't a useful parameter anymore because there are two populations: alternating and non-alternating knots, which have their own behaviours. Complex knots often have more in common with other knots of a certain class (e.g. twist, torus knots) than with the same crossing number.

More complex knots require more rope to tie.
My idea was to use this trend to figure out or constrain the topology in my experiments. For example, if my knot has 2 microns of DNA in it, I can figure out (hypothetically) that it must have at most 30 crossings based on extrapolating the ropelength curve or Bao's data, or something like that. I still haven't solved the topology-constraining problem. However, looking into the ropelength scaling got me interested in other aspects of maximally tight knots, even though maximally tight knots don't really occur in my experiments.

In addition to the lengths of ideal knots, I was interested in their size in general, which for a polymer is typically quantified by the radius of gyration, and a lot of polymer physics looks at the scaling of the radius of gyration with length. It's established that loose floppy knots grow more slowly than loose floppy rings, but what about tight knots? It turns out, the radii of gyration for ideal knots hasn't been calculated (at least not to my knowledge), so to examine this I'd have to generate the knots based on tables of 100 Fourier coefficients and then discretize and measure them. Instead I opted to look at a less-ideal system, minimal knots on cubic lattices. These were tabulated by Andrew Rechnitzer for a paper on the topic, and exist for knots up to 10 crossings (for which there are 165). These are useful for doing simulations of knotted polymers because they allow the initialization of complex knots, and the lack of tabulation for more complex knots makes them difficult to systematically simulate.

A minimal 10-crossing knot on a cubic lattice.
These are the knots whose coordinates on a cubic lattice have the least number of points, but they are still longer than ideal knots. How much longer? The length of each minimal lattice knot (which I call the cubelength) is roughly 40% greater than the equivalent ideal knot, decreasing slightly as a function of crossing number. This makes handwaving sense, because you have an extra factor of square root of two if you go across a diagonal instead of taking two steps. We can see all the different knots (from 3 to 10 crossings) in the graph below; notice there is overlap between the largest 7-crossing knots and the smallest 10-crossing knots. I had a hypothesis that these tight knotted lattice polygons would be more compact in their scaling (the radius of gyration would increase with a weaker power of length) than regular compact lattice walks, and went to examine it.
Every minimal lattice knot up to 10 crossings (except one of the 10-knots...I only ended up with 164 for some reason).

The size of a random walk grows with the square root or 0.5 power of its length. The size of a self-avoiding walk grows faster, with roughly the 0.6 power, for reasons I explain here. A compact globule (the form a polymer takes when it collapses) which grows by wrapping around itself, will grow with a volume proportional to its length, so its radius is expected to grow with the 0.33 power of its length. What do we see when we look at lattice knots? The ensemble of compact lattice knots grows at the 0.33+-0.02 power, entirely consistent with a compact globule.

Left: Average izes of minimal cubic lattice knots as a function of the crossing number. Right, same as a function of their length. I intuitively guessed the function Rg=(L/6)$^{1/3}$ and it fit the data really well.
Nothing too exciting there, but another question one could ask is whether minimal knots grow faster or slower than random self-attracted walks of comparable length, as polymers scale differently as they grow, before they reach the asymptotic limit. To figure this out, I generated self-avoiding random walks on a cubic lattice up to length 60, biasing the steps they take (with a Metropolis method) to have a strong preference for being next to non-adjacent occupied sites. Even with very strong self-attraction (well past the theta-point in polymer language), the scaling was 0.36, and not 0.33, but looking at a graph there is not a noticeable difference between the scaling of knots and tight lattice walks.

So, what's the moral of this story? A tangent from my research into DNA knots lead me to look into some of the scaling properties of ideal and minimal cubic lattice knots. I didn't find anything too profound: more complex ideal knots are bigger, and minimal cubic knots grow like the compact globules they are. It was a fun little investigation that I don't think anyone had bothered doing before. I didn't find anything interesting enough to publish, so I thought I'd share it here.