The Rims are there to help you shape the contact patch on the ground.
The width of the rims determines the angle that the sidewalls make to the tread and thus how much the tyres deform when being used. Slicks have weak sidewalls held 'stiff' by air pressure alone.
If you have sidewalls at right angles to the tread, then the kart can move around without disturbing the tread on the track. On the other hand this movement makes the handling slightly less predictable.
If you have the sidewalls at an obtuse angle (ie the rim is wider than the tread) the tyre is stiffer but also less able to absorb any roll in the kart, either because you are turning or because the track isn't level. So you get less contact patch on the ground.
What you have to do is balance between track conditions where you want the maximum amount of tread on the ground and live with the kart moving around on the tyres (narrow rims, lower pressures) and conditions where the predictability of handling is more important.
For most drivers, predictability of handling is key, however in theory, if the track is low grip then you would move to narrower rims, because you are already sliding around and the extra grip is more important than the movement on the tyres.
The other thing is that your driving style will make a difference. So changing the rims may balance (or unbalance) the kart as far as grip (front to back) is concerned. This may be easier than changing your driving style!
So as always :- test, test, test but be aware that the results may help you on one track on one day, but when track conditions are different the same rime may make you go slower.
Wet tyres are built with a steep (almost right angle) between the tread and sidewall to give you the ability to move round on the tyre while keeping the contact patch on the road. They also have stiffer sidewalls to stop the side distorting so much
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