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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.

enter image description here

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

enter image description here

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

enter image description here

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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).

enter image description here

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.

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    $\begingroup$ Could you give us a sense of scale -- where is the unit circle in your picture? I'm assuming horizontal is the direction of the real axis. Also, what are the other colours? I see red and green, other than blue. $\endgroup$ Jun 9, 2021 at 17:45
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    $\begingroup$ Thanks, that's a nice image. Do you know how the picture changes if you plot the roots of all symmetric polynomials w/integer coefficients and $p(1)=1$? i.e. restrict to the simplest class of polynomials that contains the Alexander polynomials? This should be significantly simpler work than what you have already done. $\endgroup$ Jun 9, 2021 at 19:57
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    $\begingroup$ For the achiral points, have you checked to see if there is anything interesting that distinguishes the strong achiral points (of various types) from the weak achiral points? If there is anything interesting going on with that, it might point you in a direction. $\endgroup$ Jun 10, 2021 at 19:07
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    $\begingroup$ Are you aware of the restrictions on the Alexander polynomials of achiral knots? doi.org/10.1007/BF01420117 See also mathematik.uni-regensburg.de/friedl/papers/… These criteria indicate that it might be helpful to further distinguish the Alexander polynomials of - and + achiral knots in your plots. However, I don't think necessary and sufficient conditions have been obtained. $\endgroup$
    – Ian Agol
    Jun 12, 2021 at 5:27
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    $\begingroup$ @RyanBudney Look at theorem 3.1 in the paper, he doesn’t assume strong amphichirality. $\endgroup$
    – Ian Agol
    Jun 12, 2021 at 13:48

2 Answers 2

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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.

enter image description here

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.

enter image description here

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.

enter image description here

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  • $\begingroup$ Your first picture somehow reminds me of an electromagnetic field experiments with iron filings. $\endgroup$
    – user277578
    Jun 11, 2021 at 18:03
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    $\begingroup$ Some of that presumably is Moire Effect -- the cartesian grid coming from how I generate the polynomials vs. the cartesian grid of pixels on the screen. This is me repeating a comment of Jon Hillman, who has read the thread. $\endgroup$ Jun 11, 2021 at 18:59
  • $\begingroup$ @RyanBudney Just a heads up: it’s moiré, not Moiré. It’s not a name. $\endgroup$ Jun 11, 2021 at 20:41
  • $\begingroup$ All typos belong to me, and do not reflect on Jon Hillman. $\endgroup$ Jun 11, 2021 at 21:08
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    $\begingroup$ I just want to say that these are beautiful and fascinating images, thank you for making them! $\endgroup$ Jun 11, 2021 at 23:29
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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.

Dan Christensen polynomials with coefficients bounded by 4

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.

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