Karp reduction is polynomial time computable many-one reduction between two computational problems. Many Karp reductions are actually one-one functions. This raises the question whether every Karp reduction is injective (one-one function).
Is there a natural $NP$-complete problem that is known to be complete only under many-one Karp reduction and not known to be complete under injective Karp reduction? What do we gain (and lose) if we define $NP$-completeness using injective Karp reduction?
One obvious gain is that sparse sets can not be complete under injective Karp reductions.