Unknotting Knot Theory

We’ve all been there - you take your headphones out of your bag, and they’ve tied themselves into a labyrinth of loops and twists. Perhaps you encounter a stubborn shoelace knot that just won’t come undone. Surprisingly, nature faces the same set of problems. For instance, DNA molecules loop and knot together inside our cells, leaving scientists to figure out how to untangle them without cutting through. Understanding these tangled loops is the focus of an entire branch of mathematics called Knot Theory. This field studies how loops twist, turn, and knot in three-dimensional space, and helps us answer a fundamental question: when is a knot truly a knot? How can we tell if two tangled knots are unique or simply variations of the same loop?

Knot theory is a branch of topology:  the study of shapes that remain unchanged despite deformations like bending, twisting and stretching. A mathematical knot in this context is defined as a non-self-intersecting closed curve: a loop that never crosses itself. 

Knot Theory dates back to the idea given by Carl Friedrich Gauss, however, the modern theory stems from suggestions given by William Thomson, a British mathematician and physicist also known as Lord Kelvin. Kelvin proposed that atoms might be made of tiny knots of vortex tubes of ether. To investigate this idea, mathematicians like Peter Guthrie Tait began classifying and cataloging different knots. While Kelvin’s “knot atom” theory was later disproven, the groundwork evolved into the modern knot theory. By the 20th century, tools like knot polynomials allowed researchers to distinguish knots with remarkable precision.

Two knots can look completely different at first glance but may actually be the same once you start moving them around. In knot theory, what matters is not the drawing on the page, but the shape of the loop in space. If one knot can be stretched, twisted, or flipped until it matches the other, without ever cutting the loop, they are considered identical. It might be possible to simplify a knot drawn with ten crisscrossing strands to something much neater, while a simple-looking tangle might be impossible to untie. This makes cataloging knots challenging, as a new tangle might just be a twist of an existing one. 

Since knots can look very different yet still turn out to be the same, mathematicians required ways to tell them apart with certainty. To do this, they used something called invariants: features of a knot that never change, no matter how much you stretch or twist it. The Jones polynomial is one such tool, which assigns a unique polynomial to each knot. It was discovered by Vaughan Jones in 1984 and is an invariant of an oriented knot or link, meaning it remains unchanged under certain transformations of the knot. For instance, the Jones polynomial for the trefoil knot is:

V(t) = t + t³- t⁴

Tricolorability is another such property of a knot that involves coloring the strands of a knot diagram with three colors with the following rules:

1. At least two colors must be used.

2. At each crossing, the three incident strands must either all be the same color or all different colors.

This provides a simple, visual way to identify and distinguish knots.

Knot theory may have started as a mathematical idea, but today it has real-world applications in biology, chemistry, and physics, like understanding DNA and protein folding or designing new materials. What began as the study of loops and tangles has become a field that reveals the hidden structure behind both the natural world and abstract mathematics. 

Works Cited

1. Osserman, Robert. "Knot Theory". Encyclopedia Britannica, 2 Jul. 2024, https://www.britannica.com/science/knot-theory.

2. “Unknot.” Wolfram MathWorld, mathworld.wolfram.com/Unknot.html.

3. “Tricolorability.” Wikipedia, 6 Nov. 2023, en.wikipedia.org/wiki/Tricolorability.

4. Jones, Vaughan. The Jones Polynomial. 2005. https://math.berkeley.edu/~vfr/jones.pdf

5. “YouTube.” www.youtube.com, www.youtube.com/embed/8DBhTXM_Br4.