Make a lot of shallow passes - you will kill yourself and/or break blades if you try to go too deep in a given pass.
Once you've got your line scored maybe 1/4 the way through, put a board under the biggest part, just off the line, so you can push the smaller part down and open up the cut. That makes cutting deeper and finally all the way through easier and possible.
______________________________ The CoTH CYA - please consult w/your veterinarian under any and all circumstances. - ET
Tried the power tools; the razor blade works better. We did a large area. We set them on plywood over saw horses (less time on your knees). Align the cut near the edge, so you can bend open the cut. A couple strokes with the box cutter blade and the mats cut like butter. When they don't, change the blade.
One thing we did learn during that project, the mats expand and contract with heat. If you cut them in the warm sun, they will be a quarter inch smaller when you place them in the cool barn.
We have used the circular saw as well and it works great with nice clean edge and cuts in no time. The trick is to have someone slightly pull the mat away from the saw as its cutting to help prevent the saw blade from getting stuck. We easily cut mats in a fraction of the time it would take to use box cutters.
White Lennox utility knife sold in Home Depot-the kind that is really easy to change the blade, and the handle is large and comfortable. Snap a chalk line where you want the cut. Put a 2x4 on edge under the cut.
Don't push down hard. The problem most people have is that they bear down on the knife, which just pushes the rubber in on the sides of the blade making it MANY times more difficult. Keep a sharp blade in it, and find the light pressure that makes the best cut. you have to make several passes, but by having it lifted over the 2x4, it self opens away from the blade. If it does not self-open, you are pushing down too hard or the blade is dull. I've never bothered with any kind of lubricant, but it certainly wouldn't hurt. It goes really quickly and easily once you get a feel for the right pressure on the blade. Also, if you are having to pull really hard on the knife, it becomes really dangerous. If you get a helper to walk both feet down both sides of the cut, it opens up and you get a much deeper cut per pull. Do it properly, and the sides just fall away from the cut with light pressure from two fingers on the knife. Bear down with all your might, and you will be lucky to make any cut at all, other than into your leg.
If you don't have a strong enough grip to move them around easily, get a couple of C-clamps that you hand fits well in.
I cut several and used a box knife and water, dipping the knife in water not pouring it on the mat. Used a long level (because it was metal) and just did several easy passes with the level as the guide on top. I was dreading the job but was done in no time. Used one blade for six mats, the water made all the difference.