## Eight-crossing molecular knot

It was an item on NPR's Morning Edition, January 12, 2017: "Scientists Have Twisted Molecules Into The Tightest Knot Ever." Nell Greenfieldboyce was reporting on an article due out the next day in Science: "Braiding a molecular knot with eight crossings" by David Leigh and his team at Manchester. "The first molecular knot was created by chemist Jean-Pierre Sauvage, one of three scientists who won last year's Nobel Prize in chemistry for work in creating parts for future molecular machines. His knot had loops that made it look a bit like a three-leaf clover." That was back in 1989; the 3-crossing trefoil knot is in fact the simplest knot possible. Greenfieldboyce quotes Leigh: "and then for the next 25 years, chemists weren't able to make any more-complicated knots than that." Now things have changed: "But in just the past few years, scientists including Leigh have managed to produce a few more complex knots. His team's latest knot is the most intricate yet. It looks a lot like a Celtic knot and is designed to effectively tie itself in a test tube. Molecular strands wrap around metal ions that act like knitting needles and set up strand crossings in just the right spots." This knot is the one labelled $8_{19}$ in Dale Rolfsen's knot table. The Science article mentions that the molecular backbone of the knot is only 192 atoms long; as Greenfieldboyce reports, "500 times smaller than a red blood cell."

The next-to-last step in realizing the knot $8_{19}$ as a molecule. The four iron ions that serve as scaffolding are still in place in this image. Image courtesy of the Leigh group, University of Manchester.

The knot $8_{19}$ is better known as the torus knot T(3,4): it can be drawn on the surface of a torus so that it circles the the diameter 3 times and the meridian 4.

$8_{19}$ drawn on the surface of a torus. Image courtesy of the Leigh group.

This announcement was also covered in The New Scientist, Science News, Forbes, etc. More images of molecular knots, etc., are available on the Leigh group's website.

## Probabilistic inference in monkeys

"Intuitive probabilistic inference in capuchin monkeys" appeared online in Animal Cognition, October 15, 2016. The authors (Emma Tecwyn, Stephanie Denison, Emily Messer, Daphna Buchsbaum) explain in their introduction: "Extensive research has revealed that basic numerical abilities are evolutionarily ancient ... . One specific aspect of numerical cognition that has been much less studied in animals is the ability to reason about probabilities or make probabilistic inferences. The key distinction between this ability and other types of numerical competence is that reasoning about probabilities involves reasoning about relative quantities, or proportions (e.g. in a population consisting of two types of item, the quantity of one type of item relative to the total quantity of both types of item) as opposed to simple comparisons of absolute quantities."

The four reported experiments. In each experiment, the tester would pick one item from each jar with each hand. The monkey could see what was in the jars, but not what the tester had selected. The monkey would then tap on one of the tester's hands and get what it held. Capuchins prefer peanuts to monkey pellets. In describing the results of the experiments, the authors use "Jar A" for the jar (shown on the left in this figure but placed left or right at random in practice) with the best proportion of peanuts.

Results of experiments: the proportion of trials in which the monkey chose the hand with the item from "Jar A" was better than chance in all four experiments, although only "marginally" better in Experiment 4. Error bars represent one standard deviation. $n$ is the number of capuchins who participated in the experiment. Some were excluded because of "side bias" (e.g. always chose left) and some "did not participate due to a lack of motivation to come into the testing cubicles for sufficient sessions to complete the experiment." Images adapted from Animal Cognition.

From the Discussion: "In conclusion, we found evidence that at least some capuchins, like human infants and apes, were able to make inferences about single-item samples randomly drawn from heterogeneous populations (Experiment 2), and this was achieved by reasoning about relative as opposed to absolute frequencies of preferred and non-preferred items within populations (Experiment 3). This is the first evidence for intuitive probabilistic inference in a monkey species, suggesting that the ability to reason about probabilities may be evolutionarily ancient."

## Music and math on La Repubblica-TV

La Repubblica has a video series "Rep@Conference" dedicated to acquainting students in Italian schools with "protagonists in culture, sport and current affairs." On January 19, 2017 they invited Piergiorgio Odifreddi, a logician specializing in recursion theory who has become a well-known public intellectual, to talk about mathematics and music with a group of high-school students from a Liceo scientifico in Rome. The meeting is available online. It's all in Italian, of course, but it's a nice example of how a national newspaper can try to "humanize" science, and mathematics in particular. Odifreddi seems casual but is tightly organized; he concentrates on tuning and well-tempering, with the inevitability of irrational numbers, and on the way geometric symmetries can appear in music. At the end, when students ask him what his favorite number his, it's 1729 (the "Hardy-Ramanujan" number; he makes it seem almost obvious). And when he finds out that the Liceo is named after Archimedes, he has to tell the students about Archimedes' "greatest invention:" the soccer ball (i.e. the truncated icosahedron, one of the Archimedean solids): "a small step for a man, a great step for sport."

Tony Phillips
Stony Brook University
tony at math.sunysb.edu