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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?

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    $\begingroup$ 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)." $\endgroup$
    – Did
    May 29, 2015 at 18:06
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    $\begingroup$ 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. $\endgroup$ May 30, 2015 at 19:36

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We call invariants that have distinct values for each knot complete. The Jones polynomial is not complete. This is the in book Knot Theory and Its Applications By Kunio Murasugi. enter image description here

Hope this helps.

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  • $\begingroup$ What is the Kan in parentheses in reference to? Or rather, which Kan is it a reference to? $\endgroup$ Jun 17, 2015 at 17:12
  • $\begingroup$ @SeanTilson Here is the reference from the book: [Kan] T. Kanenobu, Examples on polynomial invariants of knots and links, Math. Ann.275 (1986) 555-572. And the link to the text is maths.ed.ac.uk/~aar/papers/murasug3.pdf $\endgroup$
    – N. Owad
    Jun 18, 2015 at 4:14
  • $\begingroup$ Is there a way to know what the genus of the knots are? I'm just trying a super naive approach to find some example of knots with identical jones polynomials and different genus $\endgroup$ Oct 14, 2023 at 1:35

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