The enzyme DNA gyrase produces negative supercoils in DNA. The "negative" refers to the directionality of the supercoil, not the electrical charge of the molecule. Reverse gyrase produces positive supercoils, meaning the DNA supercoils in the opposite direction of the negative supercoil.

DNA gyrase makes it twist

Gyrase does, which relives torque after Helicase breaks the Hydrogen bonds.

An aminocoumarin is any of a class of antibiotics which act by inhibition of the DNA gyrase enzyme which is involved in bacterial cell division.