What Is Knot Theory? Mathematics of Tangles That Cannot Be Undone
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A mathematical knot is a closed loop in space, and the question is whether two tangled loops can be deformed into each other without cutting. That sounds like a puzzle and turns out to connect to physics, chemistry and the behaviour of DNA.
What the objects are
A knot in this sense is a closed curve in three-dimensional space, formed by taking a knotted string and joining the ends so that the knot cannot be pulled out. Two knots count as the same if one can be deformed into the other by continuous movement without passing the strand through itself and without cutting, which is the equivalence the whole subject is built on. The simplest knot is the plain circle, called the unknot, and the question of whether a complicated-looking loop is secretly the unknot is the basic problem of the field and is genuinely difficult, since a loop can be tangled in a way that requires temporarily making it look worse before it simplifies. Links are the extension to several loops that may be entangled with each other.
How knots are distinguished
Proving two knots are different requires finding something that cannot change under deformation:
- •Crossing number, the fewest crossings in any diagram of the knot, which is easy to define and hard to compute
- •Tricolourability, a simple test using three colours that distinguishes the trefoil from the unknot immediately
- •Knot polynomials, algebraic expressions computed from a diagram that are unchanged by deformation
- •The Jones polynomial, discovered in 1984, which detected differences earlier methods missed and connected the field to physics
- •Knot groups, derived from the space around the knot rather than from the knot itself
- •Hyperbolic volume, a geometric quantity that distinguishes a great many knots
Where it came from
The subject began from a physical theory that turned out to be wrong. In the 1860s Kelvin proposed that atoms were knotted vortices in an invisible medium filling space, with different elements corresponding to different knots, which would have made a catalogue of knots a catalogue of the elements. Tait began compiling such a catalogue, tabulating knots by crossing number, and produced the first systematic tables. The atomic theory was abandoned within a few decades as the medium it depended on was disproved, and the mathematics survived and developed independently. That sequence, where a physical hypothesis generates a mathematical field that outlives it entirely, is not unusual, and the tables Tait compiled by hand are still the basis of how knots are catalogued.
The unknotting problem
Deciding whether a given tangled loop is actually the unknot is the central computational question and its difficulty is genuinely surprising. A loop can require being made more complicated before it simplifies, so no method that only ever reduces crossings will always succeed, and diagrams exist that need substantially more crossings during unknotting than they started with. The problem is known to be decidable, meaning an algorithm exists that always terminates with the right answer, and the known algorithms are impractically slow for large diagrams. Its precise complexity is unsettled, with the problem known to sit in a class that suggests it may be easier than the hardest problems and no efficient method having been found. That combination, an obviously stated question about a piece of string with an unresolved computational status, is characteristic of the field.
Where it turns up
The subject acquired applications long after it stopped being about atoms. DNA forms knots and links during replication and must be unknotted for a cell to divide, which is done by enzymes that cut a strand, pass another through and rejoin it, and knot theory supplies the tools to work out what those enzymes did from the products they leave, which has been used to determine their mechanisms. Synthetic chemistry has produced molecules tied into specific knots deliberately, and their properties differ from unknotted equivalents. Physics connections run deep, with the Jones polynomial linked to quantum field theory in work that won a Fields Medal, and knotted structures appearing in fluid dynamics and in proposals for fault-tolerant quantum computing. Practical knot tying is a separate subject and the mathematics says little about it, since real knots have ends.
The takeaway
A closed loop in space, with two counting as the same if one deforms into the other without cutting, and deciding whether a tangled loop is secretly a circle is genuinely hard. Invariants including polynomials distinguish knots by producing something deformation cannot change. The field began from a wrong theory of atoms as knotted vortices and now describes how enzymes untangle DNA.