Here's something most people have never considered: how you tie your shoes might actually matter. Derek Muller makes this claim in his Veritasium piece, noting that one method — going counterclockwise around the loop versus clockwise — produces a fundamentally more stable knot. It's a small observation with massive implications, because it points to something happening all around us that we rarely think about: the mathematics of knots isn't just abstract curiosity. It's foundational to how our biology works.
Knot Theory's Real-World Impact
Muller writes, "This is a whole branch of mathematics that aims to identify, categorize and understand every possible knot that could ever exist." That's a striking claim — and it gets stranger. The field has apparently discovered over 352 million distinct knots. Not thought experiments or theoretical possibilities: actual mathematical objects, catalogued. This isn't pure abstraction either. Muller notes the applications are everywhere. "It is core to the structure of proteins and DNA," he writes, "it's leading to new materials that could be stronger than Kevlar" — and it's being used to develop medicines that save millions of lives.
This is where the piece becomes genuinely useful for readers who assumed knot theory was a curiosity for mathematicians in tower. The practical applications aren't hypothetical — they're actively shaping medical research and materials science right now.
The Central Problem: How Do You Tell Knots Apart?
The heart of Muller's coverage is what he calls the "knot equivalence problem" — essentially, how do you prove two knots are actually different? He writes that this question "is so famously difficult that it's propelled the entire field of knot theory for over 150 years." The most famous hook comes from Alan Turing's final publication: "No systematic method is yet known by which one can tell whether two knots are the same... a decision problem which might well be unsolvable is the one concerning knots."
This is where Muller builds real tension. He walks through the history — Kelvin's vortex theory of atoms, the Scottish physicist Peter Guthrie Tate's smoke machine experiments that produced those famous rings, the way these mathematical objects were supposed to explain atomic composition. Then he shows how the theory collapsed under the weight of actual experimental evidence.
The historical narrative is effective because it makes the mathematics feel alive. But Muller also demonstrates where current research stands: algorithms exist to distinguish knots from each other, though the upper bound on moves required is astronomical — larger than the number of stars in the observable universe for even simple crossings.
Invariants and Colorability
The most accessible part of the piece involves what mathematicians call "invariants" — properties that never change no matter how much you twist or tangle a knot. Muller explains tri-colorability clearly: take a diagram of a knot, color each segment separated by crossings with at least two colors, and at intersections either all three strands share one color or they all differ. The rule is simple, but the implications are powerful.
"Tri-colorability only has two rules: first you must use at least two colors because you can color any knot in with one color and second at crossings the three intersecting strands must either be all the same color or all different colors."
This works as a test. The unknot is not tri-colorable — you can't use at least two colors without violating the rules. But the trefoil is easily tri-colorable, coloring each of the three segments a different color. Since every possible projection of these knots maintains this property, we know they're definitively different.
Critics might note that this invariant only gives you two categories across all knots — essentially "is it tri-colorable or isn't it" — which means it's not very discriminating. But Muller builds on this by showing how P-colorability expands the test to any prime number besides two, making it far more powerful for distinguishing knots like the figure-eight knot from the unknot.
Bottom Line
Muller's strongest move is connecting the abstract mathematics of Reidemeister moves — the three types of transformations that can move between projections — to real-world applications in biology and medicine. His biggest vulnerability is underestimating how difficult it actually is to tabulate knots; the sheer number (352 million) is staggering, but the difficulty of proving equivalence means we may not have found everything we're looking for. The piece succeeds when it makes knot theory feel urgent rather than academic.