Knot Invariants
Topology through diagrams, moves, and polynomials
Trefoil 3₁
Figure-Eight 4₁
Torus T(2,5)
Unknot
Hopf Link
Tube radius:
8
Rotation speed:
1.0
3D Knot Diagram
Gauss Code & Crossings
Knot name:
Trefoil 3₁
Crossing number:
3
Unknotting number:
1
Alexander Δ(t):
-t + 3 - t⁻¹
Jones V(t):
-t⁻⁴ + t⁻³ + t⁻¹
Determinant:
3
Signature:
-2
Knot invariants are quantities preserved under ambient isotopy (Reidemeister moves I, II, III). The Jones polynomial V(t) was discovered in 1984 and connects to statistical mechanics (Kauffman bracket) and Chern-Simons field theory.