Is there any genus one knot other than $5_2$ with Alexander polynomial $2t^2-3t+2$? 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 $5_2$?
 A: 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.

A: For the prime knot example, the first in the tables is $15n43522$ with diagram:
 
A: I finally joined MO this evening to ask an entirely unrelated question, but I thought I might respond to this one as well. The Alexander polynomial does not detect $5_2$, as noted in the previous answers. However, we now know that both knot Floer homology and Khovanov homology detect this knot, as shown in my recent paper with Steven Sivek:
Floer homology and non-fibered knot detection, https://arxiv.org/abs/2208.03307.
Since knotMJ mentioned the hyperbolic knot $15n43522$, I'll point out that we also showed that knot Floer homology detects membership in the set
$\{15n43522, Wh^-(T_{2,3},2)\}$
where $Wh^-(T_{2,3},2)$ is the negatively-clasped 2-twisted Whitehead double of the right-handed trefoil. Knot Floer homology doesn't distinguish these two knots. It does, however, detect the positively-clasped 2-twisted Whitehead double of the right-handed trefoil $Wh^+(T_{2,3},2)$, as we prove in the paper.
These results follow from a classification of genus-1 knots in the 3-sphere whose knot Floer homology is 2-dimensional in the top Alexander grading (we call these knots nearly-fibered, as having 1-dimensional Floer homology in the top Alexander grading implies that the knot is fibered, by work of Ghiggini and Ni).
A: Maybe a different approach but having the same flavor: Take a (untwisted) Whitehead double of any nontrivial knot $K$. Denote such a double knot $WD(K)$. Construct a satellite knot using $5_2$ knot as the pattern and $WD(K)$ as the companion. Then the Alexander polynomial of the satellite knot is the product of the Alexander polynomials of $5_2$ knot and $WD(K)$. The latter has the trivial Alexander polynomial.
