▁lead ▁to ▁the ▁same ▁value . ▁The ▁guarantee ▁comes ▁from ▁the ▁fact ▁that ▁ ▁counts ▁something , ▁independently ▁of ▁the ▁rec urrence . ▁In ▁particular , ▁ ▁gives ▁the ▁number ▁of ▁a cyc lic ▁orient ations . ▁ ▁Jones ▁polynomial ▁ ▁Al ong ▁the ▁hyper bol a ▁, ▁the ▁T ut te ▁polynomial ▁of ▁a ▁plan ar ▁graph ▁special ises ▁to ▁the ▁Jones ▁polynomial ▁of ▁an ▁associated ▁altern ating ▁kn ot . ▁ ▁Ind ividual ▁points ▁ ▁( 2 , 1 ) ▁ ▁counts ▁the ▁number ▁of ▁for ests , ▁i . e ., ▁the ▁number ▁of ▁a cyc lic ▁edge ▁subsets . ▁ ▁( 1 , 1 ) ▁ ▁counts ▁the ▁number ▁of ▁sp anning ▁for ests ▁( edge ▁subsets ▁without ▁cycles ▁and ▁the ▁same ▁number ▁of ▁connected ▁components ▁as ▁G ). ▁ ▁If ▁the ▁graph ▁is ▁connected , ▁ ▁counts ▁the ▁number ▁of ▁sp anning ▁trees . ▁ ▁( 1 , 2 ) ▁ ▁counts ▁the ▁number ▁of ▁sp anning ▁sub graph s ▁( edge ▁subsets ▁with ▁the ▁same ▁number ▁of ▁connected ▁components ▁as ▁G ). ▁ ▁( 2 , 0 ) ▁ ▁counts ▁the ▁number ▁of ▁a cyc lic ▁orient ations ▁of ▁G . ▁ ▁( 0 , 2 ) ▁ ▁counts ▁the ▁number ▁of ▁strongly ▁connected ▁orient ations ▁of ▁G . ▁ ▁( 2 , 2 ) ▁ ▁is ▁the ▁number ▁ ▁where ▁ ▁is ▁the ▁number ▁of ▁edges ▁of ▁graph ▁G . ▁ ▁( 0 , − 2 ) ▁If ▁G ▁is ▁a ▁ 4 - regular ▁graph