Ouch. You might want to have that looked at ASAP. If not, flood with dilute betadine or H2O2 (but not too long), rinse with dh20 or saline, dry, Neosporin and put a couple of butterflies on it and cover with a clean bandage. Any sign of infection, go to the doc.
A bit late for following my advice, but there is always next time...
Yeah, I would have done something like that too. Peroxide is ok, betadyne is great, but plain old soap and water work fine too. The neosporin is also great, but in my experience, it prevents the skin from binding together quickly.
If you are going to skip the stitches that are obviously needed, you ought to do this for any cut you get: wash it good, betadyne & peroxide are fine if you got 'em. Then pat nice and dry, and close the wound with butterfly closures. THEY DON'T HOLD LONG IF YOUR SKIN IS WET, or kept wet with any ointment!
Then, having gotten a nice clean, dry closure, hit the wound with a few drops of superglue as well. If it is a jagged cut, then you really ought to run down to the doctor and get it fixed up right. (yours looked fine)
Then put a really good absorbent bandage over it, and then don't bend that knee! If the wound keeps getting ripped open, you will have a huge scar and a very slow healing time. At the very best. You also have a much greater risk of acquiring an infection that can eventually turn out very badly. 200 years ago, folks died all the time from minor cuts like that.
After care should include daily inspections. A little red around the edges is to be expected, but any significant swelling, redness spreading from the injury, or a purulent discharge? Trot on down to the doctor and get ready for the stitches, antibiotics, and tongue lashing that you deserve for not patronizing the doctor sooner.
After 2 days or so, applications of neomycin may accelerate the healing, but they also soften the scab holding the skin together and encourage your non-stitched wound to fall apart.
I gigged myself below the knee a few years back with the spur on my Silky pole saw, and it was deep and about 1.25" wide. I did the home treatment on that, and it was looking great until I put on the neomycin too soon. The butterfly stitches didn't hold, and it ended up healing rather slowly. I'd have been fixed up and done with it in 10 days, had I gotten the stitches. I had no infection, and I did just fine without wasting any coin or time on a doctor.
If you are a tree worker of any sort, you tend to be the kind of person that takes risks anyway. What is just one more cut?
If you have a long history of not getting infections, I think I might have done without the stitches, too. Conversely, if your cuts tend heal slow and you get infections all the time, you ought to go see the doctor right away. After about 6 hours of delay, they generally do what is called "debridement", which essentially means that they enlarge your cut and trim out any damaged/dirty/exposed tissues. And then they put in stitches or staples. Staples are faster, cheaper, and sometimes heal quicker. They also hurt like hell when you bump them, and I think that would be a real problem right on the bend of a knee.