▁associated ▁to ▁sl n ▁for ▁all ▁n . ▁In ▁ 2 0 0 3 , ▁C ath arina ▁St ro ppel ▁extended ▁Kh ovan ov ▁hom ology ▁to ▁an ▁invariant ▁of ▁tang les ▁( a ▁categor ified ▁version ▁of ▁Res het ik hin - T ura ev ▁invari ants ) ▁which ▁also ▁general izes ▁to ▁sl n ▁for ▁all ▁n . ▁ ▁Paul ▁Se idel ▁and ▁Ivan ▁Smith ▁have ▁constructed ▁a ▁sing ly ▁grad ed ▁kn ot ▁hom ology ▁theory ▁using ▁L agr ang ian ▁intersection ▁Flo er ▁hom ology , ▁which ▁they ▁conject ure ▁to ▁be ▁is omorphic ▁to ▁a ▁sing ly ▁grad ed ▁version ▁of ▁Kh ovan ov ▁hom ology . ▁C ip rian ▁Man oles cu ▁has ▁since ▁simplified ▁their ▁construction ▁and ▁shown ▁how ▁to ▁recover ▁the ▁Jones ▁polynomial ▁from ▁the ▁chain ▁complex ▁underlying ▁his ▁version ▁of ▁the ▁Se idel - Smith ▁invariant . ▁ ▁The ▁relation ▁to ▁link ▁( k not ) ▁polynomials ▁At ▁International ▁Congress ▁of ▁Mathemat icians ▁in ▁ 2 0 0 6 ▁Mik h ail ▁Kh ovan ov ▁provided ▁the ▁following ▁explanation ▁for ▁the ▁relation ▁to ▁kn ot ▁polynomials ▁from ▁the ▁view ▁point ▁of ▁Kh ovan ov ▁hom ology . ▁The ▁ske in ▁relation ▁for ▁three ▁links ▁ ▁and ▁ ▁is ▁described ▁as ▁ ▁Sub st itut ing ▁ ▁leads ▁to ▁a ▁link ▁polynomial ▁invariant ▁, ▁normal ized ▁so ▁that ▁▁ ▁For ▁ ▁the ▁polynomial ▁ ▁can ▁be ▁interpreted ▁via ▁the ▁representation ▁theory ▁of ▁quantum ▁group ▁ ▁and ▁ ▁via ▁that ▁of ▁the ▁quantum ▁Lie ▁super