I'm wondering if its ever possible to find two non-isotopic knots which have identical jones polynomials but different seifert genus?
Attempting to google for this I found this example of non-isotopic knots with identical jones polynomials. But in the first page of the paper its clearly stated that all the knots have identical genus.
The Conway knot and Kinoshita-Teraska knot are mutants, so they share the same Jones polynomial. But the Conway knot has genus 3 and the Kinoshita-Terasaka knot has genus 2.