Knots as closed loops in 3D, projected to 2D with crossings. Each knot has topological invariants — the Jones polynomial distinguishes many (but not all) knots from the unknot.
Crossing #
3
Writhe
+3
Unknotting #
1
Alexander poly
t⁻¹ − 1 + t
Chirality
Chiral (left)
V(t) = −t⁻⁴ + t⁻³ + t⁻¹
Reidemeister moves:
The trefoil knot is the simplest non-trivial knot. It cannot be deformed into its mirror image (chiral). Jones polynomial V(t) = −t⁻⁴ + t⁻³ + t⁻¹ distinguishes it from the unknot.
Jones polynomial (1984): defined via Kauffman bracket, invariant under Reidemeister moves II and III. One of the great surprises: quantum field theory (Chern-Simons) gives a natural definition.