Traditionally, PIE phonology postulates three voiceless velar/uvular stops to PIE:
*/ḱ/ (c), */k/ (q), */kʷ/ (q̆)
But I made a search for a PIE dictionary with come 11-15 thousand words, and found only one occurrence of the combination ku̯, in word kroku̯a̯ "border post" which is highly untypical, while kʷ is abundant.kʷ
Given this fact, the only explanation that comes to mind is the idea that there was actually no kʷ in PIE but rather in most cases it was a combination ku̯. This theory would be refuted if there were many kʷ in the end of the roots: in that case after applying this theory we would have roots ending with -ku̯ which is prohibited. So I searched for -ekʷ- composition throughout the roots so to check if it actually frequent.
And what I found?
In many cases either the u̯ in -eku̯- can be part of a suffix, or the *kʷ can be actually reconstructed as *p!
soq̆os "sap" - may be actually, sopos
teq̆os "running water" - may be actually, tequ̯os (suffix -u̯-)
a̯eq̆ea̯ "water body" - may be actually a̯epea̯ (this root a̯ep- for water is actually reconstructed along a̯eq̆ea̯)
u̯lq̆os "wolf" could be actually u̯lquos (suffix), or u̯lpos or lupos
coq̆r "excrement" could be copr (compare "coprophagy")
i̯eq̆r "liver" could be i̯epr (compare "hepatitis")
o̯oq̆ "eye" could be o̯oqu̯-, a typical u-stem (while -q̆ stem is highly untypical)
peq̆ter "cook" could be pepter (compare "peptide").
If this is wrong, what could be the solution to the problem?
A similar solution that the voiced equivalent to q̆, ğ at the end of the roots, could be actually b may explain why PIE has very rare *b. For example, traditionally reconstructed e̯reğos "darkness" could be actually e̯rebos which literally coincides with Greek word for "darkness", "erebos"!
See also this question What evidence supports labialized velars in PIE?
Note that the traditional phonology implies that in several unrelated branches such as Hellenic, Osco-Umbrian and Celtic, happened similar change:
which seems to me highly unlikely. It also postulates
kʷ->p->f in Germanic, but not root-initially!
penkʷe -> finf
u̯lkʷos -> wolf
kʷod -> what
kʷor -> where
This supports my idea that root-initially kʷ could be actually ku̯ but root-finally it was p because ku̯ root-finally is impossible due to sonority rule (u̯ being sonorant cannot be further from the root nucleus than stop k)