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I had asked this question on math.stackexchange 2 days back but came up empty handed so I wanted to ask it here.

Are there known examples of $2$ non equivalent knots that have identical jones polynomials but different Seifert Genus? Specifically the strong genus which is the minima of all surfaces as opposed to the weak genus which is just the minima over all surfaces generated by Seifert's algorithm (although honestly an counter example with weak genus is welcome as well)

Besides being a cool example to think about I was also simply curious if the Jones Polynomial has been "experimentally shown" to detect Seifert Genus (i.e. we have no counter examples so far).

The link in the math.stackexchange question lists knots which have same jones polynomial and same genus but are non equivalent.

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    $\begingroup$ The Conway knot and Kinoshita-Terasaka knot are mutants, so have the same Jones polynomial, but have different Seifert genus. $\endgroup$
    – Ian Agol
    Commented Oct 16, 2023 at 15:48
  • $\begingroup$ That’s very interesting! They look so similar yet the genus varies anyways. Do you want to post that as an answer so I can accept? $\endgroup$
    – Sidharth Ghoshal
    Commented Oct 16, 2023 at 15:56
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    $\begingroup$ I’ve answered the question on math stackexchange since this is not really a research-level question. $\endgroup$
    – Ian Agol
    Commented Oct 17, 2023 at 2:33
  • $\begingroup$ excellent thank you! $\endgroup$
    – Sidharth Ghoshal
    Commented Oct 17, 2023 at 2:37

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