Concept
When can two knots be deformed into each other without cutting? From Kelvin's atom theory to DNA biology.
Understand it in one breath
"Can one tied loop be moved into another without cutting it?" Kelvin's 19th-century atomic model failed, but the classification problem endured. Invariants such as the Jones polynomial distinguish many knots, yet no such polynomial distinguishes every pair. Knot theory now connects with DNA topology and quantum field theory.
At a glance
Key formula
Key moments
Kelvin — knots as atoms
William Thomson, later Lord Kelvin, proposed that atoms were knots in a pervasive ether. The physical hypothesis failed, but the effort to classify knots endured.
The Alexander polynomial
James Alexander assigned a polynomial invariant to a knot, making it possible to prove algebraically that many apparently similar knots are different.
Knots observed in circular DNA
Electron microscopy revealed knots in circular DNA. Later work used topology and knot theory to study how topoisomerase enzymes create and resolve knots and links in DNA.
The Jones polynomial — a Fields Medal breakthrough
Vaughan Jones discovered a new knot polynomial with surprising links to statistical mechanics and quantum field theory. The work contributed to his 1990 Fields Medal.
Modern applications
DNA replication and knot-resolving enzymes, topological qubits, protein folding, and synthetic chemistry.
Beyond MathVoyage
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