Jones polynomials:

1. The original one via von Neumann algebras by Jones.

2. An elementary proof by Kauffman.

3. Via Temperley-Leib algebras by Kauffman.

4. Quantum group representations by Reshetikhin & Turaev.

5. Several approaches in the HOMFLY paper.

6. Via Chern-Simons theory by Witten.

7. Khovanov homology.

Proofs that Jones polynomials are invariants of links:

1. The original one via von Neumann algebras by Jones.

2. An elementary proof by Kauffman.

3. Via Temperley-Leib algebras by Kauffman.

4. Quantum group representations by Reshetikhin & Turaev.

5. Several approaches in the HOMFLY paper.

6. Via Chern-Simons theory by Witten.

7. Khovanov homology.