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Salem numbers and Lehmer's minimum height problem are venerated not only in number theory and diophantine analysis, where they are considered naturally interesting for their own sake, but also in fields such as hyperbolic geometry and holomorphic dynamics. As is so well known, the least known Salem number is a root $1.176280\ldots$ of the following monic reciprocal $10$-th degree $\{-1,0,1\}-$polynomial discovered way back in 1933 by D. H. Lehmer in his work on primality testing: $$ x^{10} + x^9 - x^7 - x^6 - x^5 - x^4 - x^3 + x + 1. $$

I am interested in seeing any mathematical contexts or computations through which this particular polynomial shows up, apparently accidental occurrences included (not to say preferred).

Here is an example from topology: this is the Alexander polynomial of infinitely many knots, including the $(-2,3,7)$-pretzel knot.

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Here's a paper of McMullen where the Lehmer polynomial shows up (see Theorem 1.2 there):

http://www.math.harvard.edu/~ctm/papers/home/text/papers/blowup/blowup.pdf

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    $\begingroup$ Here are the slides for a talk of McMullen's math.harvard.edu/~ctm/expositions/home/text/talks/princeton/… which cover that example and many others. $\endgroup$
    – David E Speyer
    Commented Sep 4, 2013 at 16:30
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    $\begingroup$ Also this is an example in dynamics, and so maybe is not quite what you were looking for. In any case, this is an excellent question, and I look forward to the answers. $\endgroup$
    – Lucia
    Commented Sep 4, 2013 at 16:42
  • $\begingroup$ It is a beautiful article. In this connexion there are two additional papers by McMullen: "Coxeter groups, Salem numbers and the Hilbert metric," and "Cyclotomic factors of Coxeter polynomials." (The latter is joint with B. Gross and Eriko Hironaka.) Luckily, also available is the full video of McMullen's 1 hour lecture: youtube.com/watch?v=lbd4WEckLWs . $\endgroup$
    – Vesselin Dimitrov
    Commented Sep 4, 2013 at 20:28
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well, this link will direct you to just under 300 scholarly articles on Lehmer's polynomial:

http://scholar.google.com/scholar?q=polynomial+lehmer&btnG=&hl=en&as_sdt=2005&sciodt=0%2C5&cites=18047953845360790641&scipsc=1

(oh, and I added the big-list tag to this question, I guess that is appropriate :)

a small selection:

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    $\begingroup$ Thank you for this search. The first paper, by Eriko Hironaka, is particular interesting. We learn from her article that the first historical appearance of that polynomial actually appears to be in Reidemeister's book on knot theory, which was published in 1932 and preceded Lehmer's paper by several months. A remarkable coincidence. $\endgroup$
    – Vesselin Dimitrov
    Commented Sep 4, 2013 at 20:33

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