Zeroes of the Alexander polynomial for achiral knots Are there some known properties about the position (on the complex plane) of roots of the Alexander polynomial of achiral knots? They are shown as blue points in the following picture of roots for knots up to 16 crossings. Specifically what about those sitting in the middle of the holes around chiral ones.

Added the zoom-out picture: Zeros (in the half unit disc) of the Alexander polynomial for knots up to 16 crossings (red - chiral,alternating; green - chiral,nonalternating; blue - achiral) blue on top of red on top of green

Edit: A fragment of the roots for the positive amphichiral knots (red dots):

------------------- Edit:
In the picture below we indicate (by orange dots) the roots (of multiplicity more than 1) of Alexander polynomials of prime knots up to 15 crossings (313.230 knots, regardless of their chirality). The number shows the maximal multiplicity among all polynomials.
As suggested by this picture, the holes and their centers have much to do with the multiplicity of roots after all. As in the suggested paper by Hartley, for positive amphicheiral knots, the roots are more frequently have higher (than one) multiplicity because they "potentially" factors through the other polynomials (at least to the second power).

The isolated roots in the interior are from Alexander polynomials like: $(t^4-3t^3+5t^2-3t+1)^2$, $(t^6-2t^5+4t^4-5t^3+4t^2-2t+1)^2$, $(t^6-5t^5+12t^4-15t^3+12t^2-5t+1)^2$, $(t^4-5t^3+9t^2-5t+1)^2$, $(2t^4-6t^3+9t^2-6t+2)^2$, $(2t^4-7t^3+11t^2-7t+2)^2$, $(t^6-3t^5+5t^4-5t^3+5t^2-3t+1)^2$ ... so they are squares of Alexander polynomials for some knots.
 A: This is an extended comment, and a response to one of my earlier comments.
The image below depicts the roots of all integer-coefficient Laurent polynomials
$$p(t) = \sum_i a_i t^i$$
that satisfy $p(t^{-1}) = p(t)$, $p(1)=1$ that are of degree at most 6 with coefficients at most 6.  i.e. the sum is over $i=-6 \cdots 6$ and the coefficients $a_i \in \{-6, \cdots, 6\}$.  These latter limitations were to help make the computation reasonably-long.
As in your figure, I only show the roots inside the unit disc with non-negative imaginary part.

I think the main difference between your image and this one must come down to the manner in which the polynomials are generated.
I believe you are getting Alexander polynomials of much higher degree, but also your coefficients are not uniformly distributed like in the way I generate the polynomials.
So I think your plot is very much a reflection of the "shape" of Alexander polynomials, parametrized by the crossing number.  As you have observed, it appears certain regions in the space of possible Alexander polynomials are more rapidly filled-in by the amphichiral knots.
Perhaps there is a construction that can produce low-crossing amphichiral knots that can also produce a sequence of "approximating" chiral knots, where "approximating" is in terms of roots of the Alexander polynomial.
If you choose the polynomials to be at most degree 26, with coefficients between -2 and 2, uniformly and randomly you get something even further from your plot.

I tried a less even-handed search through Alexander polynomials, biased towards polynomials that are weakly alternating, like the typical polynomials one sees in knot tables.  Interestingly (maybe only to me), the answer appears to be even further from what you are getting.  Clearly the phenomena I see in the tables isn't enough to characterize what is happening.

A: As you point out in the modified question, positive achiral knots  have Alexander polynomials that tend to factor by Hartley’s result. So the roots will be roots of smaller degree polynomials.
When you plot roots of polynomials with relatively small coefficients (as in Ryan Budney’s answer), the roots of higher degree polynomials tend to avoid the roots of smaller degree polynomials. Dan Christensen plotted roots of polynomials with coefficients in $[-4,4]$, colored by the polynomial degree, where one sees this phenomenon: roots of lower degree polynomials tend to be isolated.

This seems to explain your picture heuristically: the roots of Alexander polynomials of achiral knots tend to be isolated because they are the roots of lower degree polynomials.
