What genre are groups like Chevelle, Helmet, Tool

As papakias and Johnny Bones correctly point out these bands cover many genres eg Soundgarden - Grunge/Heavy Rock/Metal. Slipknot - Metal. Rise Against - Punk. System of a Down - Metal. Seether - Grunge. Three Days Grace - Rock. Nine Inch Nails - Industrial/Techno. Tool - Prog.

I do disagree however on these bands being under the Rock/Metal or Metal genres though. Some of these bands contain no elements that could be considered Metal (specifically the modern day interpretation of Metal in which Led Zeppelin and Deep Purple, originally known as Metal bands are now labeled Classic Rock or Heavy Rock).

So I would have to say Heavy Rock as a blanket genre that would cover these diverse bands.

These are different genres of music. Tool and A Perfect Circle are progressive metal/rock, sum 41 are punk etc. If I had to chose one common root/genre for all of them I would say rock/metal.