Leonard Kelley holds a bachelor's in physics with a minor in mathematics. He loves the academic world and strives to constantly explore it.
Anyone who has tied a great knot and needs to unravel it will attest to the complexity of what initially seems a simple object. From tying your shoes to basic seamanship, knots come in a wide variety yet somehow have patterns to them. How can we unravel them? And in doing so, what will we stumble upon that will totally surprise us? The science of knots is fascinating, but don’t get too twisted up as we explore on.
What knot is the best one for a given situation? Humans have determined for various situations different knots that best establish what works, but oftentimes it’s though trial-and-error. Can mathematics offer us the ability to pick a knot with given attributes that is maximally beneficial for our desired outcome? Work by Khalid Jawed (MIT) might be giving us just that. Part of the challenge is in the different ways forces play out in the arrangement of the material, and with essentially many point-places of forces happening, developing a map of any given knot is tough. So we start simple, and Jawed’s group first eliminated high coefficients of friction by working with metal wires made up of nitonol (“a hyper-elastic nickel-titanium alloy”) for their knots. Specifically, one of the simplest knots known as the trefoil (which involves us putting one end of our wire though subsequently created loops). By holding down one end of the wire and measuring the force needed to complete each braid, the researchers found that as the number of twists increased, the force required to complete the knot grew too, but at a greater-than linear rate, for 10 twists needed 1000 times the force of a single twist. It’s a first step towards a mathematical landscape for knot theory (Choi “Equation”).
Why is it that when we look at knitted materials, they have different properties that their constituents don’t? For example, most base elements used are not elastic and yet the knitted material is. It all boils down to the patterns we use, and for Elisabetta Matsumoto (Georgia Institute of Technology) that means coding the properties of the base slip-knots to show the meta-level attributes we see as emergent behavior. In another study by Frederic Lechenault, it was demonstrated how properties of the knitted fabric could be determined by the “bendiness” of the material, how long it is, and “how many crossing points are in each stitch.” These contribute to the conversion of energy that can happen as the material is stretched, with subsequent rows pulling at the slip knots and therefore deflecting energy around, allowing stretching and eventual return to the rest state possible (Ouellette).
As most of us will attest, sometimes we get something so tangled up that we would rather toss it than deal with the frustration of unraveling the knot. So imagine scientists’ surprise when they found a class of knots that will undo themselves – no matter their state! Work by Paul Sutcliffe (Durham University) and Fabian Maucher looked at vortices that were tangled, which seems the same as knotted but implies a seeming lack of order. That is, one could not look at a tangle and be easily able to reconstruct the stages of how it got there. Of course, you could undo the tangle by cutting and stitching together, but the team instead looked at the electrical activity of a heart that often gets tangled. They found that no matter what they looked at, the electrical tangles undid themselves, but how it was done remains a mystery (Choi “Physicists”).
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Knots in Fluids?
We associate knots with string-like objects, but scientists have found evidence that knots can be found in other places also. Shocking, often seemingly impossible places like…fluids? Yes, evidence points to water, air, and other fluids having knots potentially being the key to deciphering the mystery of turbulence. Ideas of this began with Lord Kelvin in the 1860s and evolved over time but the essential reasoning for why knots even appear in the first place or how they change are quite mysterious still. For example, fluids with no viscosity will retain their total knottedness, but no one knows why. Experimentation would be great but generating knots in fluids for study has been a challenge in itself to establish. Work by William Irvine (University of Chicago) has possibly shed some insight but uses hydrofoils (objects that help to displace water) to finally create a vortex knot to study. Randy Kamien (University of Pennsylvania) used lasers on liquid crystals. These works may also apply to electromagnetic fields, too (Wolchover).
Choi, Charles Q. “Equation Works Out Kinks in Knot Math.” Insidescience.com. American Institute of Physics, 09 Oct. 2015. Web. 14 Aug. 2019.
---. “Physicists Surprised to Discover Knots That Can Escape Complex Tangles.” Insidescience.com. American Institute of Physics, 19 Jul. 2016. Web. 14 Aug. 2019.
Ouellette, Jennifer. “Physicists are decoding math-y secrets of knitting to make bespoke materials.” Arstehcnica.com. Conte Nast., 08 Mar. 2019. Web. 14 Aug. 2019.
Wolchover, Natalie. “Could Knots Unravel Mysteries of Fluid Flow?” quantamagazine.org. Quanta, 09 Dec. 2013. Web. 14 Aug. 2019.
This content is accurate and true to the best of the author’s knowledge and is not meant to substitute for formal and individualized advice from a qualified professional.
© 2020 Leonard Kelley