ov , ▁then ▁at ▁the ▁University ▁of ▁California , ▁Davis , ▁now ▁at ▁Columbia ▁University . ▁ ▁Over view ▁To ▁any ▁link ▁diagram ▁D ▁representing ▁a ▁link ▁L , ▁we ▁assign ▁the ▁Kh ovan ov ▁bra cket ▁[ D ], ▁a ▁chain ▁complex ▁of ▁grad ed ▁vector ▁spaces . ▁This ▁is ▁the ▁analog ue ▁of ▁the ▁K auff man ▁bra cket ▁in ▁the ▁construction ▁of ▁the ▁Jones ▁polynomial . ▁Next , ▁we ▁normal ise ▁[ D ] ▁by ▁a ▁series ▁of ▁degree ▁sh ifts ▁( in ▁the ▁grad ed ▁vector ▁spaces ) ▁and ▁height ▁sh ifts ▁( in ▁the ▁chain ▁complex ) ▁to ▁obtain ▁a ▁new ▁chain ▁complex ▁C ( D ). ▁The ▁hom ology ▁of ▁this ▁chain ▁complex ▁turns ▁out ▁to ▁be ▁an ▁invariant ▁of ▁L , ▁and ▁its ▁grad ed ▁E uler ▁characteristic ▁is ▁the ▁Jones ▁polynomial ▁of ▁L . ▁ ▁Definition ▁This ▁definition ▁follows ▁the ▁formal ism ▁given ▁in ▁D ror ▁Bar - N atan ' s ▁ 2 0 0 2 ▁paper . ▁ ▁Let ▁{ l } ▁denote ▁the ▁degree ▁shift ▁operation ▁on ▁grad ed ▁vector ▁spaces — that ▁is , ▁the ▁hom ogeneous ▁component ▁in ▁dimension ▁m ▁is ▁shift ed ▁up ▁to ▁dimension   m   +   l . ▁ ▁Similarly , ▁let ▁[ s ] ▁denote ▁the ▁height ▁shift ▁operation ▁on ▁chain ▁complex es — that ▁is , ▁the ▁r th ▁vector ▁space ▁or ▁module ▁in ▁the ▁complex ▁is ▁shift ed ▁along ▁to ▁the ▁( r   +   s ) th ▁place , ▁with