Have Been Cleaning With Alcohol And Bandaging A Thumb Cut With A Box Cutter. Is It Infected As It Has Become Numb And Painful Now?

I cut my thumb a week ago with a box cutter. Thought I could have used stitches but opted not to. Cut was long enough and deep enough were stitches would have been ideal. I been cleaning with alcohol and bandaging it properly. But not its numb and very hard and painful at the site. It's also tingling. Is it infected. Also at the tip it's kind of dark not black but just bruised dark.