The colouring rule
Use three colours. At every crossing, the over-strand and the two under-strand arcs on either side must either all be the same colour or all be different. If you can colour the entire knot this way using all three colours, the knot is tricolourable. The trefoil is; the unknot is not — and no deformation can ever change that.
What is a topological invariant?
A topological invariant is a property that does not change under any continuous deformation — stretching, bending, twisting — as long as you do not cut or pass strands through each other. Tricolourability is one such invariant: if a knot is tricolourable, every diagram of it is tricolourable, no matter how tangled the drawing.
This is the same concept at work in the other polytopo submenus: the Euler characteristic χ = V − E + F is an invariant of surfaces; the ℤ/2 parity of a rotation is an invariant of loops in SO(3). Invariants are how topology classifies the world.
Where knot theory sits in the topology landscape
In Edges & Surfaces, Euler counted edges and faces to produce an invariant of surfaces (χ). In Twists & Spinors, the ℤ/2 parity of a rotation is an invariant of loops in rotation space. Knot theory is the third pillar: it produces invariants of loops embedded in ordinary 3D space. All three are instances of the same deep programme — algebraic topology — which assigns algebraic objects (numbers, groups, polynomials) to geometric shapes so that equal algebra implies equivalent shape.
The trefoil's tricolourability is a mod-3 invariant. But knot invariants grow far more powerful: the Jones polynomial (1984, Fields Medal) assigns a polynomial in a variable t to each knot, distinguishing knots that all earlier invariants missed. It was discovered not through geometry but through connections with statistical mechanics and quantum field theory — the same physics that underlies the spinor story in Twists & Spinors.
DNA topology — topoisomerases unknot DNA strands during replication. A knotted chromosome cannot be copied; the cell dies. Many cancer chemotherapies target topoisomerase II directly — trapping the enzyme mid-cut leaves lethal knots in tumour DNA.
Protein folding — roughly 1% of known proteins contain a knotted backbone. Knot type correlates with mechanical stability: knotted proteins resist unfolding under tension, relevant in force-bearing cell-wall enzymes.
Topological quantum computing — fault-tolerant qubits can be encoded in anyons whose world-lines braid around each other. The computation is stored in the knot type of these world-lines: knot invariants become the computation itself. This is the direct descendant of the spinor and rotation story in Twists & Spinors, now operating in the language of knots.
