Joe. Normally i clean my cuts out with thinner and super glue them here in my shop. Cant tell after a coulpe days there was a cut. I couldnt get this one to stop bleeding also had very little movement from the knuckle forward!! Still cant move it much but I sure can feel it!!!
Did they do x-rays by any chance? I once cut the top of My thumb at the first joint, was similar to yours but maybe a little worse, and the blade was a brand new pruning blade, the curved kind that cuts really well, and was thicker, so It basically left a nasty strip in the middle and cut on each side of the blade. Yeah, a freaking hand saw, one swipe, and did that much damage. I never went to the ER even though I probably should have, but I could not stop working at the time, and still had to use It, so I flushed with alcohol, closed It with steri strips, and wrapped It up the best I could.. It kept trying to heal up, then splitting open no matter what I tried. it was just in a bad place. Anyhow the reason I ask is because I actually cut into the bone of the knuckle, and later could easily feel the bone missing. Eventually It healed up, but It stayed stiff for many years, and I could not close that thumb. Even to this day It's not the same, and I get some bad arthritis in that joint. You may have cut deeper then You realize. Mine was obvious because It spread out like a banana peel, and could see right to the bone, although I didn't realize until later that It got that far into My knuckle.
A band saw can do some real damage, looks like you caught It just in time before things got really ugly. If It got into the bone though, then possible you will be feeling it for a while, and It will be hard to close that finger. For me It wasn't pain that stopped It from closing. It just did not have that much range of motion for some reason. Hope It heals quickly for ya.
Greg
