Animated 3D knot curves with crossing structure — knot group and polynomial invariants
Knot theory studies embeddings of circles in 3D space up to continuous deformation (isotopy). The crossing number is the minimum crossings in any diagram. Invariants like the Jones polynomial (1984) and HOMFLY polynomial distinguish non-equivalent knots. The trefoil is chiral (left- and right-handed versions are distinct). The figure-eight is amphichiral. Torus knots T(p,q) wind p times in one direction and q in another on a torus surface. Open problem: does the Jones polynomial detect the unknot?