Maybe instead of a knife amnesty, the authorities could try an amnesty for thugs, drug dealers, drunk drivers, fraudsters, muggers, wife beaters, child abusers, football hooligans and sex offenders. Now that would really reduce crime but I'm sure the police and the politicians would say its too difficult.
I don't think there is any evidence population density makes any difference to crime apart from crimes of opportunity and a confusion over the intepretation of statistics. When I was travelling by London underground in the rush hour, it was very crowded but it never cross my mind to attack another passenger. If the assumption was true, then the highest crime rates would be in rush hour trains, rock concerts and major sporting events.
Similarly when I was out in the country, I did not cross my mind to attack anyone either. Big cities tend to have a lot of crime so some people automatically assume the denser the population, the more likely the crime without considering other factors like if the type of people living in big cities is different from those living in smaller cities or in rural places.