Cut thumb near nail over month ago. thumb still hurts at time. cut not completely closed tight.. iis there possibility of infection? no swelling

See your doctor: Many things are possible, including permanent damage to the nail bed, to the underlying bone and to the attached ligaments. If the wound has not healed in a month, you need to see a doctor. Absence of swelling is a reasonable indication of the lack of an infection, but a soft tissue only wound should have completely healed in 2-3 weeks at age 51.