Knot theory is the branch of mathematics that studies closed loops in three-dimensional space that are tangled and cannot be untangled without cutting. A mathematical knot is a closed curve embedded in 3D space; two knots are considered equivalent (isotopic) if one can be continuously deformed into the other without the curve passing through itself. The simplest knot is the unknot (a plain circle); the simplest non-trivial knots are the trefoil knot and the figure-eight knot.
Knot invariants are mathematical quantities assigned to knots that remain unchanged under continuous deformation — they allow us to distinguish non-equivalent knots. Important invariants include the crossing number (minimum number of crossings in any diagram), the knot group (the fundamental group of the knot's complement), the Alexander polynomial, the Jones polynomial (discovered by Vaughan Jones in 1984, earning him the Fields Medal), and the HOMFLY polynomial. Reidemeister showed that two knot diagrams represent the same knot if and only if they are connected by a sequence of three simple moves (Reidemeister moves).
Knot theory has surprising connections to physics and biology. In physics, knot invariants (Wilson loops) appear in quantum field theory and topological quantum computing, where anyonic excitations can be manipulated by braiding to perform computations fault-tolerantly. In biology, enzymes called topoisomerases change the topology of DNA — which forms knots and supercoils during replication — by cutting and rejoining strands, a process that can be modelled using knot theory.
Mathematicians compute knot invariants — numbers or polynomials assigned to each knot that remain unchanged under any deformation. If two knots have different invariants, they must be different. Common invariants include the crossing number, the Alexander polynomial, and the Jones polynomial. No single invariant distinguishes all knots, but together they provide powerful classification tools.
Reidemeister moves are three local transformations on knot diagrams that do not change the underlying knot: Type I (twisting/untwisting a loop), Type II (sliding one strand over another), and Type III (moving a strand past a crossing). A theorem by Kurt Reidemeister states that two knot diagrams represent the same knot if and only if one can be converted to the other by a sequence of these three moves.
The Jones polynomial is a knot invariant discovered by Vaughan Jones in 1984 using operator algebras related to quantum mechanics. It is a Laurent polynomial in a variable t that takes different values for topologically distinct knots. It can distinguish many knots that the classical Alexander polynomial cannot, and its discovery revealed deep connections between knot theory, quantum groups, and statistical mechanics.
DNA in cells forms knots and supercoils during replication and transcription. Enzymes called topoisomerases manage these topological changes: type I topoisomerases cut one strand to relieve supercoiling; type II topoisomerases pass one double-stranded segment through another, changing the knot type. Understanding these operations mathematically helps biologists study how cells replicate DNA without tangling it permanently.
Topological quantum computing stores quantum information in non-abelian anyons — exotic quasi-particles in certain 2D quantum systems. Logical operations are performed by braiding anyons around each other, which corresponds to changing the topology of their worldlines (a form of knot operation). The key advantage is fault tolerance: topological quantum information is protected from local perturbations because it depends only on the global topology of the braid, not on precise physical details.