Geometric Topology – Genus One Knot with Alexander Polynomial 2t^2-3t+2

Question

It is known that genus one fibred knots are two trefoils and the figure-eight knot. Is there any characterization of the knot $5_2$? Specifically, is there any other genus one knot that shares the same Alexander polynomial $2t^2-3t+2$ with $5_2$?

Answer 1

Ian Agol, in the comments says:

Yes, there should be plenty. Think of the Seifert surface for the 5_2 knot as a disk with two strips (1-handles) attached. By tying knots into the strips (with zero framing so as not to change the linking form), you can obtain many knots with a genus 1 Sefert surface with the same Seifert form and hence same Alexander polynomial.