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Knot Theory Explained: From Shoelaces to DNA and the Jones Polynomial | Veritasium
Derek Muller introduces knot theory by contrasting simple loops with real knots, then guides you through how mathematicians classify and distinguish knots. He covers the unknot and first nontrivial knots like the trefoil, explains the knot equivalence problem, and shows how tools such as Reidemeister moves and a growing set of invariants help prove when two knots are different. The video links these ideas to powerful polynomials, historical tabulations, and surprising applications in biology, chemistry and materials science, including DNA topology and synthetic molecular knots. All of this centers on understanding how a seemingly humble knot can unlock a vast landscape of theory and real-world impact.
Introduction to Knots and the Core Question
Knot theory begins with the idea that a knot is a closed loop, and the simplest knot is the unknot, a plain circle. The first nontrivial knot is the trefoil, and distinguishing knots from each other without cutting the string is the knot equivalence problem. Kurt Reidemeister later showed that three simple moves are enough to transform any two diagrams of the same knot into each other, but proving two knots are different requires more than just trying every move. This section sets up why knot theory matters and how mathematicians approach the problem.
"You only need three types of moves to transform any two identical knots into each other the twist, the poke, and the slide." - Derek Muller, Veritasium
Invariants and What They Do
As knots twist and turn, certain properties remain unchanged. The crossing number captures the minimal number of crossings, while tricolorability and colorability in general provide coarse but powerful distinctions. These invariants are the fingerprints that help distinguish knots across different projections, and they form the backbone of practical knot classification even before more sophisticated tools are used.
"The Jones polynomial is like the Alexander, but with the more specific equation for the second rule that lets it distinguish many more knots." - Derek Muller, Veritasium
From Invariants to Polynomials
Beyond simple counts and colorings, knot theory employs polynomials, starting with the Alexander polynomial, which was the standard for decades. The field was revolutionized by Vaughan Jones, whose polynomial invariant added the power to distinguish many knots that the Alexander polynomial could not separate. The Homfly polynomial followed, offering yet another refinement. These invariants work together like multiple identifiers against a backdrop of Reidemeister moves, enabling researchers to differentiate knots more effectively than with any single invariant alone.
"Knot theory is used to analyze the knots they tie or untie and how they operate as a result." - Derek Muller, Veritasium
History, Tabulation, and Modern Computation
The video traces the long arc from early hand-drawn knot tables to modern computer-aided catalogs. From Tate, Kirkman and Little’s 19th-century work to the 20th-century realizations of Reidemeister moves and today’s large-scale tabulations, the story shows how collaborative effort and computational power have expanded our knowledge. The modern era includes impressive milestones such as the completion of prime knots up to high crossing numbers, driven by algorithms that use invariants to filter duplicates and verify counts across vast datasets.
"DNA increases its stiffness by super coiling, and you can do the same with your wires." - Derek Muller, Veritasium
Real-World Connections: Biology, Chemistry, and Materials
Knots are not just abstract curiosities. In biology, knotting occurs in DNA and proteins, influencing replication and folding. Enzymes such as type 2 topoisomerases untangle DNA to enable replication, while knots in proteins affect function. Chemists have even created synthetic knotted molecules, pushing toward materials that combine strength and novel properties. Knot theory thus informs understanding at the molecular scale and inspires the design of new materials with real-world impact, including durability and biological interactions.