To cut is to break and to separate. This dream can be the allegorical representation of your circumstances in real life: it is perhaps a broken relationship or the end of a pattern of behavior. (See KNIFE) If someone else cuts you, a friend will betray you.
Improving memory can be done in a number of ways, and one of those is by having a good night's sleep. During sleep, your mind is still busy and going through the process of consolidation, where it strengthen your memories as well as the "practice" skills, which you have learned while you're awake. According to the associate professor at NYU Langone Medical Center, Dr. Rapoport says that something happens while we are all sleeping, and it helps us learn things better. Thus, if you're trying something new, you can perform it better right after you sleep.
Reading in dream
Sometimes you will not be able to differentiate lucid dream. If you want to be sure, try reading a simple sentence, if you are able to read then you are in wakeful stage if not you are in lucid dreaming. Studies show that you cannot read while dreaming or you will not find the repeated sentences one more time.