Well, quite a lot of conflicting advice for the OP here and IMO a lot of it is baseless nonsensical myth rather than scientifically proven fact.
First of all lets look at the one about the slick tyre having more grip than a treaded tyre. This may indeed be true for a tyre running on a smooth surface in dry conditions where grip is purely a function of friction and contact area but, in all other conditions (damp, dusty, gravel, oily, etc) the smooth tyre on a smooth surface is likely to be less than optimal. Once a lubricant is added to the equation (and water is quite a good lubricant in the case of rubber) then you need to look for other ways to maintain grip. Looking at car tyres, and grip in general, the introduction of a tread pattern takes two forms. The first is a large tread pattern designed to remove water and allow the tyre to make contact with the road surface without 'floating' on a film of lubricant (eg, water). The second form of tread is a finer pattern that provides a mechanical grip somewhere between lubricant dispersal and microscopic friction. This is the world of sipes (go on,
google it. You know you want to!) and this really comes into its own on wet surfaces with a less than perfect finish. Here the mechanical grip gained by adding a fine tread pattern far outweighs the friction lost by reducing the contact area with the removal of rubber. I think a fairly good analogy for this is to imagine you are slowly sliding down a steep rockface towards certain doom. Sure, you can try to flatten yourself against the rock to maximise contact area and friction, hopefully slowing or stopping your descent, but in reality nothing really comes close to the effectiveness of jamming your fingers and toes into any available nook, cranny or crack! This is what sipes, and fine tread patterns, do. They are not about removing water. The lumps, bumps, edges and corners they present to the road surface provide opportunities for mechanical grip where water has reduced the ability of microscopic friction to be the sole provider of grip. In essense friction and this fine mechanical grip are much the same but on a different scale. Water or lubricant can easily foil the pure friction while it has a lesser effect on the traction provided by tread pattern and sipes.
The compromise is that a treaded tyre will most likely have slightly less grip and higher rolling resistance when on a warm, dry road but given the downside that riding on wet/damp roads brings then having a treaded tyre on a mixed use, multi weather bike is a good idea.
The second issue mentioned in this thread is the relationship between tyre pressure/footprint/puncture occurances. This is purely my hypothosis and opinion but other than running a tyre underinflated to the point where pinch punctures occur then I think the relationship can be largely ignored as any differences are negligable. A tyre run at a lower pressure may indeed deform around a sharp object rather than pressing itself onto the point so as to be pierced, but when this is balanced against the fact that a tyre at a higher pressure has a smaller contact area on the road so is statistically less likely to encounter a sharp object I think it is likely to be a draw and neither high nor low tyre pressure will suffer a significantly higher number of punctures.
Finally, it doesn't matter what tyre you run, one day you will encounter that smooth, shiny, greasy piece of tarmac when you enter a turn just a bit too fast and BANG! You WILL go down, period.
And finally finally. Going back to the world of the motor car, lets take a look at what the F1 racing teams have traditionally done for tyres. If it is dry then a slick, tread free tyre is the choice. When it is damp or slightly wet then intermediates with a mild tread pattern are selected. Once it is lashing down with rain and there is lots of standing surface water then monsoon tyres with deep, water displcing tread is chosen. A treaded tyre can be run on a dry road with very little loss of performance and loss of grip is gradual and more predictable than a slick tyre which suddenly gives up gripping when pushed beyond it's limits in the wet, often catastrophic consequences. Getting back to the OP's question. I recommend getting tyres with some tread for wet weather use (or another bike with treaded tyres and guards
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