Three professors argue it is non-polar.
My professor argues that it is a monopole, like most ions.
The structure of the triiodide ion places a negative formal charge on the central iodine atom. The molecular geometry is also linear (at least according to VSEPR), and its electronic geometry is trigonal bipyramidal. Given the high amount of symmetry, shouldn't triiodide be non-polar? Sure, it might have a charged central atom, but so does carbon dioxide.
What's the answer?
The structure of the triiodide ion places a negative formal charge on the central iodine atom.
No it doesn’t. The two resonance structures that describe the four-electron three-centre bond put the negative formal charge on the outer iodines ($\ce{1/2-}$ each).
That said, polarity is usually defined as having a non-zero dipole moment. The dipole moment’s vector must display the same symmetry as the entire molecule. Since the molecule is linear and both $\ce{I\bond{...}I}$ distances are equal, its point group is $D_{\infty \mathrm{h}}$ which includes $i$ which means the overall dipole moment must be zero.
The charge it carries does not matter. Any single-atom ion also has zero dipole moment and would thus be called non-polar. ‘Non-polar’ does not mean ‘free of electrostatic interaction’.
I would argue it is a polar molecular ion.‡ In the comments to this question there has been already pointed out that the term "polarity" is quite ill-defined.
Many chemists understand that if a molecule has a vanishing dipole moment, the molecule itself is non-polar; at least this is true since molecules are neutral by definition. If you actually can extend that definition to charged species is open for debate. I personally dislike this definition - molecule or not.
In this framework carbon dioxide is non-polar, but it reacts with polar substrates, like Grignard agents. The reason for this is its quadrupole moment.
A much more general definition of the term polarity hence should include any molecule that has a multipole of any kind. We can then even include ions.
A polar molecular entity is a molecule that possesses a multipole.
Since the triiodide anion has a non-zero quadrupole moment, I would consider it polar. This point has been and will always be open for debate.
A quick DF-BP86/def2-SVP calculation determines the natural atomic charges as $q(\ce{I_{\mu}})=-0.11$ (the atom in the middle) and $q(\ce{I_{term}})=-0.44$ (the terminal two; deviations from $1.00$ due to rounding). The same calculation finds:
Traceless Quadrupole moment (field-independent basis, Debye-Ang):
XX= 10.9525 YY= 10.9525 ZZ= -21.9050
XY= 0.0000 XZ= 0.0000 YZ= 0.0000
‡: It is not a molecule, since it is not neutral, which is a prerequisite from the IUPAC goldbook definition.
If you accept the Atkins Physical Chemistry definition of polar:
A polar molecule has a permanent electric dipole moment arising from the partial charges on its atoms
Then the answer turns upon whether triiodide is actually linear.
Isolated Triiodide and Pentaiodide Anions in the Crystal Structure of [Rb(Dibenzopyridino-18-Crown-6)2]2(I3)(I5) Zeitschrift für anorganische und allgemeine Chemie Volume 634, pages 1247–1248, says it is bent with an angle of 174.8 degrees.
Theoretical Study of the Solvent Effect on Triiodide Ion in Solutions J. Phys. Chem. A 1998, 102, 2065-2071, says that triiodide is linear in the gas phase, but "The free-energy surface of the same ion in aqueous solution is distinctly different and indicates a drastically enhanced probability for structures with lower symmetry."
So in the gas phase it would be non-polar by the Atkins definition of polar, while in other situations it can be slightly bent and therefore slightly polar.
