UB knot theorist Bill Menasco. The whiteboard in the background depicts drawings of mathematical knots and surfaces that Menasco created from memory. Credit: Douglas Levere – See more at: http://www.buffalo.edu/news/releases/2016/02/017.html#sthash.1rCiO0em.dpuf
But take it from Bill Menasco, a knot theorist of 35 years: This field of mathematics, rich in aesthetic beauty and intellectual challenges, and has applications. It involves the study of mathematical knots, which differ from real-world knots in that they have no ends. Each one is a string that crosses over itself a number of times, then reconnects with itself to form a closed loop. Today, we know the study of knots could have applications in surprising areas. It could enable security firms to create better encryption systems, or elucidate the mysteries of how the body unravels DNA.
Neighbors know him by his license plate: KNOTPROF, but in the world of mathematics, Menasco is perhaps best known, with Morwen Thistlethwaite, for solving the Tait Flyping Conjecture, one of three famous problems posed in the late 1800s by Peter Guthrie Tait, a father of knot theory. “You find knots in cultures all over the world. You see them in Asian art. You see them in Celtic crests. Sailors know the figure-8 knot, which crosses over itself four times. They call it the stopper.”
“The central question is pretty much the same as when I started: How do you tell two knots apart?” Menasco says. Even when two knots look very different, it’s often possible to physically manipulate them by pulling or flipping sections of the string until the two knots are twins. (Cutting, which Menasco calls “cheating,” is not allowed.)
series of drawings of nautical knots. This conversation piece, which hangs in UB knot theorist Bill Menasco’s office, was a gift from his antique-collecting father. Knots play a prominent role in sailing culture, as well as in many other cultures around the world. “It seems to be somehow buried in our humanity that we’re interested in these things,” Menasco says. Credit: Douglas Levere – See more at: http://www.buffalo.edu/news/releases/2016/02/017.html#sthash.1rCiO0em.dpuf
Knot theory seeks to encode information about knots—including what happens when they are altered in some way—into algebraic expressions that can distinguish one knot from another. While experts have devised expressions that can differentiate some knots from others, a formula capable of describing every knot uniquely remains elusive.
APPS: Knot theory has uses in physics, biology and other fields.
Eg1. when cells divide, the DNA inside them must be replicated. This requires the DNA’s double-helix structure to unfurl, generate a 2nd copy of itself that’s intertwined with the first, and then get untangled. To achieve this, enzymes help to cut, unknot and reconnect the strands. Knot theory provides insight into how hard it is to unknot and reknot various types of DNA, shedding light on how much time it takes the enzymes.
Eg2: In security, Menasco and his graduate students are partnering with the firm SecureRF to further develop the Algebraic Eraser, a security protocol that uses knots in cryptography. The protocol weaves mathematical information from knots into encryption keys that allow users to decipher secret data encoded in security badges, mobile payment devices and more.
Do knot theorists have any insight on preventing real-life knots? Scientists have done experiments in which they tumbled strings around in a box to figure out what causes tangling. The research suggests that one solution to preventing knots is to store ear buds, phone chargers and other annoying cables in smaller boxes.
UB knot theorist Bill Menasco’s bookshelf. Credit: Douglas Levere – See more at: http://www.buffalo.edu/news/releases/2016/02/017.html#sthash.1rCiO0em.dpuf
What are the origins of knot theory?
Knots first came to prominence in the 1860s, when scientists were trying to understand the nature of matter. At the time, people didn’t know that atoms were nuclei surrounded by electrons, so Lord Kelvin proposed a different theory: He hypothesized that the basic building blocks of matter were knots in the ether, a hypothetical substance that permeated space. He postulated that every element—H, O, Au and so on—was made from a different kind of knot. “So the knots are stable: They have certain properties,” Menasco says. “You can heat them up. You can cool them down. You can do certain things to them, but gold is gold, and oxygen is oxygen.” The theory fulfilled some key requirements for a system of building blocks for matter. Eg. many types of knots could exist, comprising many elements, and knots could be linked to form molecules.
UB knot theorist Bill Menasco’s personalized license plate. Credit: Douglas Levere – See more at: http://www.buffalo.edu/news/releases/2016/02/017.html#sthash.1rCiO0em.dpuf
Peter Guthrie Tait, a Scottish physicist, began an extensive study and tabulation of knots, with the goal of understanding how to tell different knots apart from one another, Menasco says. Tait created, in essence, the basis for a periodic table of knots in which H would be the “unknot”—the unknotted circle—and heavier elements would be knots whose filaments crossed over themselves an ever-higher number of times.
But the theory was wrong… “It was,” Menasco says. “But when the physicists lost interest in Kelvin’s work, the mathematicians stepped in and said, ‘Wow.’ Tait tabulated all the knots up to those with 10 crossings, and it was an amazing feat because this table remained mostly unchanged for 100 years.”
One of the last corrections to Tait’s table happened in 1973, when Kenneth Perko discovered that two of Tait’s 10-crossing knots were actually the same. These knots became known as the Perko Pair, reducing the number of known 10-crossing knots from 166 to 165.
Menasco is writing a textbook on mathematical discoveries that resulted from a 30-plus-year research partnership with Joan Birman of Columbia University and Barnard College. He hopes to finish the book, co-authored with his former PhD student, Doug LaFountain, an assistant professor at Western Illinois University, this year.
The topic? Two-dimensional surfaces associated with knots and links, which are chains of multiple knots joined together.
The subject is difficult to explain to a non-mathematician, but here is how Menasco describes it: “Our work was on understanding surfaces associated with knot and links when they are positioned in (x,y,z)-space as a closed braid that winds around the z-axis.”
http://www.buffalo.edu/news/releases/2016/02/017.html
Recent Comments