Is there a name for the involution on Laurent polynomials? This is a simple terminology question: I want to know if the involution $z \mapsto z^{-1}$ on Laurent polynomials (over some ring, I happen to be working over $\mathbb{Z}$ but that's not important) has a special name.
My motivation is perhaps a little unusual for this site.  I'm doing some computations that involve manipulating Laurent polynomials and, being a lazy sort of fellow, I'm letting the computer do it.  Being extra lazy, I don't particularly want to learn a new programming language to do this so I'm using Perl as it's the only one that I know.  However, there my laziness stops as whilst there's a Perl module for ordinary polynomials there isn't one for Laurent polynomials.  Still, it wasn't hard to adapt it to Laurent polynomials so I did and the program is chugging away churning out these computations to its heart's content.  In writing the methods (meaning, things you can do to a Laurent polynomial), most already have obvious names (add, subtract - actually called sub_, mul(tiplication), and so forth) but I don't know one for the obvious involution $z \mapsto z^{-1}$.  inv sounds a little to easy to mistake for inverse.
So, is there a name for this?  If not, would anyone like to suggest one (preferably with an unambiguous shortening - I've already gotten fed up of typing monomial every time)?
 A: I think the answer to the original question is that there is no special name for the involution (otherwise it would have turned up by now).   My first encounter with it was in the 1979 Kazhdan-Lusztig paper on Hecke algebras, where they use a
bar notation and combine this involution on Laurent polynomials with the inversion in a given Coxeter group to get an action on the Hecke algebra of that Coxeter group.   The bar notation makes it unnecessary to invent a name for the
involution on Laurent polynomials, but "bar involution" will certainly do.
By the way, a ring of Laurent polynomials (say over $\mathbb{Z}$) provides a nice nontrivial example for a graduate algebra course, even though it rarely if ever occurs in textbooks.    It's natural to ask what are the prime ideals and factor rings, etc.  Most often the examples of commutative rings which students see are too boring and predictable to motivate the ideal machinery.    
A: I would call it the antipode.  If your base ring is commutative, then the Laurent polynomials are the coordinate ring of the multiplicative group, and the antipode gives you the inversion on the group scheme.
A: I'm with David here.  It's "the bar involution," a very hard involution to write the symbol for alone.
A: "Argument involution"? I have never heard a name for the map, and I'm sure
I would use "the $z\leftrightarrow z^{-1}$ map" as the name, in a paper.
A: Well, to offer a somewhat garbled paraphrase T.S. "Old Possum" Elliot
(see for example, http://www.americanpoems.com/poets/tseliot/5536):
You may think at first I'm as mad as a hatter
But The Naming of Things is a difficult matter,
It isn't just one of your holiday games!
Having so established my credentials in this matter, I present the following:
Think of the map $z \to z^{-1}$ as swapping charts on the Riemann sphere. (I'm OK
with here, right? I mean, I haven't looked carefully at this stuff in quite awhile.)
Think of the $z$-coordinate as corresponding to the North Pole. Then the $z^{-1}$- coordinate goes with the South Pole. 
Now, like Ben Webster, I gots out my lil' ol' wiki hammer, yes I did, but just
gave the great wiki mountain a tiny tap: what broke off was: http://en.wikipedia.org/wiki/South_Star.
The North Star is Polaris; the South Star is one Sigma Octantis.
Therefore, reminiscent, of the notion that the map $z \to z^{-1}$ is somewhat of
a swap twixt north and south, how about calling your involution something based
on Sigma Octantis? 
List:
SigmaOctantisInvolution----way to long to type
SigOct or sigoct----shorter, not too long, you'll not likely forget it
gee, what about
$\Sigma$ or perhaps $\sigma$ ----hard to do on a keyboard; BUT there is
"sigma or perhaps just "sig"---beginning to look like "real math"!
Ah ha! Abara K'Dabara--I create as I speak!  From now on, I call the involution $z \to z^{-1}$
on the algebra of Laurent polynomials $\sigma$.  You could call it sig for short; if
that's a reserved word in Perl use a variant like sigoct etc. Or perhaps even
better, for a function in a computer language, siginv--for the sigma involution,
though I'd probably just go with sig if I could.
BTW, this question inspired one of my own: https://mathoverflow.net/questions/48994/humorous-curious-unusual-names-for-mathematical-entities
HAPPY HOLIDAYS LADIES AND GENTLEMEN OF MO--HO! HO! HO!
