Before tire manufacturers started producing specialized tires, racers would run their street tires Probably the most popular tire was the Dunlop K70, which is still popular for stock British bikes (along with the round profile Dunlop K81)
Many legends rode to greatness on the Dunlop K70
You can see they were pretty popular tires with the ladies as well....
Currently Dunlop, Maxxis and Continental produce dirt track specific tires...(Goodyear was bought out by Dunlop) Dirt track tires are unidirectional and are rotated as the left side of the tire wears down.
The Dunlop "S" pattern is probably the most famous tread in flat track racing. I see this pattern when I go to sleep at night.
Look at this lucky bastard flying around on his Dunlop CD2 Dirt Track tires. The envy of those lacking in manhood. Too afraid to get on the track whilst buying every flat track shirt Lucky Brand and Johnson Motors offer.
Modern DT tires come in different compound densities from CD1 (being the softest) to CD10 (the hardest). On smoother tracks like clay you tend to run a softer compound while tracks with surfaces like pea gravel tend to demand harder tires.
As for cutting the tires, this is called "grooving". The little slices spread apart and create additional tread in the tires so they hook up better.
Ideal's heated knife (used for grooving) about $93 bones. Great if you're looking to buy a stocking stuffer for your favorite dirt tracker this Christmas. (PM me for my mailing address)
There's a whole science behind setting up your tires and every racer has their own methods of grooving, tire pressures, compounds etc...