Deep cut on finger while cooking one week ago (no stitches). finger is still swollen, red (in the area of the cut) and hand is throbbing. when hand is elevated it feels better than when hand is put down, it feels then heavy and aching. what to do?

It appears that the cut is infected and should be examined by a doctor to ascertain if systemic or local antibiotic treatment is indicated. It would be prudent to seek care soon to avoid spread of infection.

Sounds like an infection either cellulitis, or perhaps a deeper infection involving the tendon sheath or palmar space which would require drainage along with antibiotics. You need to see a hand specialist