▁ 2 . ▁The ▁Ar f ▁invariant ▁is ▁the ▁substitute , ▁in ▁characteristic ▁ 2 , ▁for ▁the ▁disc rimin ant ▁for ▁quadratic ▁forms ▁in ▁characteristic ▁not ▁ 2 . ▁Ar f ▁used ▁his ▁invariant , ▁among ▁others , ▁in ▁his ▁ende avor ▁to ▁class ify ▁quadratic ▁forms ▁in ▁characteristic ▁ 2 . ▁ ▁In ▁the ▁special ▁case ▁of ▁the ▁ 2 - element ▁field ▁F 2 ▁the ▁Ar f ▁invariant ▁can ▁be ▁described ▁as ▁the ▁element ▁of ▁F 2 ▁that ▁occurs ▁most ▁often ▁among ▁the ▁values ▁of ▁the ▁form . ▁Two ▁n ons ing ular ▁quadratic ▁forms ▁over ▁F 2 ▁are ▁is omorphic ▁if ▁and ▁only ▁if ▁they ▁have ▁the ▁same ▁dimension ▁and ▁the ▁same ▁Ar f ▁invariant . ▁This ▁fact ▁was ▁essentially ▁known ▁to ▁, ▁even ▁for ▁any ▁finite ▁field ▁of ▁characteristic ▁ 2 , ▁and ▁Ar f ▁proved ▁it ▁for ▁an ▁arbitrary ▁perfect ▁field . ▁ ▁The ▁Ar f ▁invariant ▁is ▁particularly ▁applied ▁in ▁geometric ▁topology , ▁where ▁it ▁is ▁primarily ▁used ▁to ▁define ▁an ▁invariant ▁of ▁- dimensional ▁manif olds ▁( sing ly ▁even - dimensional ▁manif olds : ▁surfaces ▁( 2 - man if olds ), ▁ 6 - man if olds , ▁ 1 0 - man if olds , ▁etc .) ▁with ▁certain ▁additional ▁structure ▁called ▁a ▁fr aming , ▁and ▁thus ▁the ▁Ar f – K erva ire ▁invariant ▁and ▁the ▁Ar f ▁invariant ▁of ▁a ▁kn ot . ▁The ▁Ar f ▁invariant ▁is ▁analog ous ▁to ▁the ▁signature ▁of ▁a ▁manifold , ▁which ▁is