# Is there a one to one correspondence between Jones' polynomials and knots?

I know Jones' polynomial is a knot invariant. By using knot invariant like p-coloration one can only say whether two knots are different but not whether they are the same. So it is like injective mappings. I was wondering whether more powerful knot invariant exists (which can tell whether two knots are same or not) and is Jones' polynomial one of them?

• mathworld.wolfram.com/JonesPolynomial.html "All prime knots with 9 or fewer crossings have distinct Jones polynomials. However, there exist distinct knots (and even knots having different crossing numbers) that share the same Jones polynomial. Examples include (05-001, 10-132), (08-008, 10-129), (08-016, 10-156), (10-025, 10-056), (10-022, 10-035), (10-041, 10-094), (10-043, 10-091), (10-059, 10-106), (10-060, 10-083), (10-071, 10-104), (10-073, 10-086), (10-081, 10-109), and (10-137, 10-155) (Jones 1987)."
– Did
May 29, 2015 at 18:06
• Due to Gordon and Luecke the knot complement is a complete knot invariant. Using a result of Waldhausen (which gives a complete invariant for knot complements) the map induced by inclusion of the boundary $\pi_1(\partial) \to \pi_1(M-K)$ is a complete knot invariant. May 30, 2015 at 19:36