Your reading it wrong. the highest is three. (which is the chest) you can see the table at the top, plus click on the "XVII" which is what I guess your looking at, its just how its labled, that is simply for three. Each item needs its own "sockets" affix, thus that are named as such, up to 13 affixs, the name does not relate to how many they give.
d3inferno
According to this data, the estimated max sockets on equiqment will be (a total of) 13-14...
http://www.d3inferno.com/mods/Socketed.html
Enjoy.
http://www.d3inferno.com/affixes/Sockets_XVII.html
for example, notice when you click it:
Mod Name Min Value Max Value
Socketed 3 3
http://wowbox.tw/d3/
had this for a very long time.
Though I wish weapons could hold my gems =/