Fix $n$, and consider the characteristic polynomials for all $C=2^{\frac{n(n-1)}{2}}$ adjacency matrices representing undirected, unweighted graphs on $n$ vertices.



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> Are the characteristic polynomials distributed somewhat uniformly among adjacency matrices? Or are some characteristic polynomials "much more popular" than others? If so, is there some Zipf-law distribution?
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The work of [Andries Brouwer](https://www.win.tue.nl/~aeb/graphs/cospectral/cospectralA.html) counts the number of characteristic polynomials for smallish $n$ relative to the number of *graphs*.




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Similarly fix $n$, and consider the Alexander polynomials associated with the $C=n!\times \lfloor{\frac{n!}{e}}\rfloor$ [grid diagrams](https://picturethismaths.wordpress.com/2015/05/03/grid-diagrams-for-knots/) that represent various links/knots.



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> Are the Alexander polynomials distributed somewhat uniformly among the grid diagrams?
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The work of [Canterralla, Chapman, and Mastin](https://arxiv.org/abs/1512.05749) suggests that a Zipf-like law tends to apply for random knot diagrams - some knot types are more popular than others. I suspect that there are more grid diagrams that have an Alexander polynomial associated with, say, $7\_6$ than with $7\_1$, because the symmetries of $7\_1$ make it harder to be randomly generated. But if $n$ were large enough, maybe uniformity takes over.




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My motivation considers a uniform superposition $\frac{1}{\sqrt{C}}|A \rangle|B \rangle$ - $A$ being over adjacency matrices (grid diagrams) and $B$ being a characteristic polynomial (Alexander polynomial) of the respective adjacency matrix (grid diagram). Over all $|A \rangle$, is $|B \rangle$ most likely uniformly distributed? If I measure $|B\rangle$ will some polynomials be more likely than others?



How are characterstic polynomials (resp. Alexander polynomials) distributed amongst adjacency matrices (resp. grid diagrams)?
The answer depends quite nontrivially on which model of random knots you consider. For a general description, see Even-Zohar's [very nice survey.](https://arxiv.org/abs/1711.10470) In many models a random knot is a very small knot (unknot, trefoil, figure eight dominating), with the resulting singularity of the distribution of Alexander polynomials.


