▁the ▁number ▁of ▁left - hand ed ▁cross ings ▁in ▁the ▁chosen ▁diagram ▁for ▁D , ▁and ▁n + ▁the ▁number ▁of ▁right - hand ed ▁cross ings . ▁ ▁The ▁Kh ovan ov ▁hom ology ▁of ▁L ▁is ▁then ▁defined ▁as ▁the ▁hom ology ▁H ( L ) ▁of ▁this ▁complex ▁C ( D ). ▁It ▁turns ▁out ▁that ▁the ▁Kh ovan ov ▁hom ology ▁is ▁indeed ▁an ▁invariant ▁of ▁L , ▁and ▁does ▁not ▁depend ▁on ▁the ▁choice ▁of ▁diagram . ▁The ▁grad ed ▁E uler ▁characteristic ▁of ▁H ( L ) ▁turns ▁out ▁to ▁be ▁the ▁Jones ▁polynomial ▁of ▁L . ▁However , ▁H ( L ) ▁has ▁been ▁shown ▁to ▁contain ▁more ▁information ▁about ▁L ▁than ▁the ▁Jones ▁polynomial , ▁but ▁the ▁exact ▁details ▁are ▁not ▁yet ▁fully ▁understood . ▁ ▁In ▁ 2 0 0 6 ▁D ror ▁Bar - N atan ▁developed ▁a ▁computer ▁program ▁to ▁calculate ▁the ▁Kh ovan ov ▁hom ology ▁( or ▁category ) ▁for ▁any ▁kn ot . ▁ ▁Rel ated ▁theories ▁One ▁of ▁the ▁most ▁interesting ▁aspects ▁of ▁Kh ovan ov ' s ▁hom ology ▁is ▁that ▁its ▁exact ▁sequences ▁are ▁formally ▁similar ▁to ▁those ▁ar ising ▁in ▁the ▁Flo er ▁hom ology ▁of ▁ 3 - man if olds . ▁Moreover , ▁it ▁has ▁been ▁used ▁to ▁produce ▁another ▁proof ▁of ▁a ▁result ▁first ▁demonstrated ▁using ▁gauge ▁theory ▁and ▁its ▁c ous ins : ▁Jacob ▁R asm ussen ' s ▁new ▁proof ▁of ▁a ▁theorem ▁of ▁Peter ▁K ron heimer ▁and