A closed loop in 3D, possibly tangled. Knots are equivalent if one can be deformed into the other without cutting. How do you tell two knots apart? Compute an invariant: a polynomial that doesn't change under deformations. The Jones polynomial (Vaughan Jones, 1984, Fields Medal) distinguishes most knots and connects to quantum statistical mechanics and topological field theory.
how a knot becomes a polynomial
A knot is an embedding of a circle into 3-space (S¹ → ℝ³). Two knots are equivalent if connected by an ambient isotopy — a continuous deformation. The fundamental question of knot theory: given two knot diagrams, are they the same knot?
Reidemeister (1927) proved that two diagrams represent the same knot iff one can be transformed to the other by a sequence of three local moves (R1, R2, R3). So invariants are quantities computed from a diagram that don't change under R1, R2, R3.
The Jones polynomial V(q) (1984): a Laurent polynomial in q^(1/2). Computed recursively via the skein relation:
q⁻¹ V(L₊) − q V(L₋) = (q^(1/2) − q^(-1/2)) V(L₀)
where L₊, L₋, L₀ are three knot diagrams differing only at one crossing (positive crossing, negative crossing, no crossing).
Examples:
• Unknot: V = 1
• Trefoil 3₁: V = -q⁻⁴ + q⁻³ + q⁻¹
• Figure-eight 4₁: V = q⁻² − q⁻¹ + 1 − q + q² (palindromic — figure-eight is amphichiral, same as mirror)
The connection to physics: Witten (1989, Fields Medal) showed Jones polynomial = partition function of Chern-Simons theory on S³ with knot as Wilson loop. Knot polynomials are quantum amplitudes. This launched topological quantum field theory and remains one of the deepest math-physics bridges.
Practical applications: protein folding (DNA and proteins form knots), supercoiled DNA, polymer chemistry, fluid vortices, and (most recently) topological quantum computers where qubits are encoded in anyon braid topology.