Noticeable Cut Under Tip After Rhinoplasty?

Keep in mind that you are only a week after surgery. All incisions go through a maturation phase during which the incision will look worse than it will once it is matured. That said, open rhinoplasty does involve an external incision and closed rhinoplasty does not. However, closed rhinoplasty may not have been indicated for you. My best advice is to get on with your life and try not to obsess about the incision. In two or three months take another look and you will be very surprised. The worse thing for you to do is to sit for hours in front of the mirror and watch your self heal. Good luck with your recovery.

The incision used for open rhinoplasty typically heals very well over time.  You can see after photos on the link below.

This scar will usually fade with time, but it may take longer than the month that your surgeon suggested. Avoiding sun exposure during this healing process also helps, as the sun can turn these fresh, healing incisions a bit dark.
In general, however, these incisions will fade very nicely, so I would recommend being patient and giving your body time to heal.