▁all ▁the ▁differential ▁maps ▁being ▁shift ed ▁accordingly . ▁ ▁Let ▁V ▁be ▁a ▁grad ed ▁vector ▁space ▁with ▁one ▁generator ▁q ▁of ▁degree ▁ 1 , ▁and ▁one ▁generator ▁q − 1 ▁of ▁degree   − 1 . ▁ ▁Now ▁take ▁an ▁arbitrary ▁diagram ▁D ▁representing ▁a ▁link ▁L . ▁The ▁ax i oms ▁for ▁the ▁Kh ovan ov ▁bra cket ▁are ▁as ▁follows : ▁ ▁[ ø ] ▁= ▁ 0 ▁→ ▁Z ▁→ ▁ 0 , ▁where ▁ ø ▁denotes ▁the ▁empty ▁link . ▁ ▁[ O ▁D ] ▁= ▁V ▁ ⊗ ▁[ D ], ▁where ▁O ▁denotes ▁an ▁un link ed ▁trivial ▁component . ▁ ▁[ D ] ▁= ▁F ( 0 ▁→ ▁[ D 0 ] ▁→ ▁[ D 1 ]{ 1 } ▁→ ▁ 0 ) ▁ ▁In ▁the ▁third ▁of ▁these , ▁F ▁denotes ▁the ▁` fl atten ing ' ▁operation , ▁where ▁a ▁single ▁complex ▁is ▁formed ▁from ▁a ▁double ▁complex ▁by ▁taking ▁direct ▁sums ▁along ▁the ▁diag on als . ▁Also , ▁D 0 ▁denotes ▁the ▁` 0 - sm o othing ' ▁of ▁a ▁chosen ▁crossing ▁in ▁D , ▁and ▁D 1 ▁denotes ▁the ▁` 1 - sm o othing ', ▁analog ously ▁to ▁the ▁ske in ▁relation ▁for ▁the ▁K auff man ▁bra cket . ▁ ▁Next , ▁we ▁construct ▁the ▁` normal ised ' ▁complex ▁C ( D ) ▁= ▁[ D ][ − n − ]{ n +   −   2 n − }, ▁where ▁n − ▁denotes