▁ 3 - man if old ▁whose ▁boundary ▁is ▁the ▁given ▁surface ▁which ▁extends ▁the ▁given ▁fr aming . ▁This ▁is ▁because ▁ ▁does ▁not ▁bound . ▁ ▁represents ▁a ▁tor us ▁ ▁with ▁a ▁trivial isation ▁on ▁both ▁gener ators ▁of ▁ ▁which ▁tw ists ▁an ▁odd ▁number ▁of ▁times . ▁The ▁key ▁fact ▁is ▁that ▁up ▁to ▁hom ot opy ▁there ▁are ▁two ▁choices ▁of ▁trivial isation ▁of ▁a ▁trivial ▁ 3 - plane ▁bundle ▁over ▁a ▁circle , ▁corresponding ▁to ▁the ▁two ▁elements ▁of ▁. ▁An ▁odd ▁number ▁of ▁tw ists , ▁known ▁as ▁the ▁Lie ▁group ▁fr aming , ▁does ▁not ▁extend ▁across ▁a ▁disc , ▁whilst ▁an ▁even ▁number ▁of ▁tw ists ▁does . ▁( Note ▁that ▁this ▁corresponds ▁to ▁putting ▁a ▁spin ▁structure ▁on ▁our ▁surface .) ▁Pon tr j agin ▁used ▁the ▁Ar f ▁invariant ▁of ▁fr amed ▁surfaces ▁to ▁compute ▁the ▁ 2 - dimensional ▁fr amed ▁c ob ord ism ▁group ▁, ▁which ▁is ▁generated ▁by ▁the ▁tor us ▁ ▁with ▁the ▁Lie ▁group ▁fr aming . ▁The ▁isomorphism ▁here ▁is ▁via ▁the ▁Pon tr j agin - Th om ▁construction . ▁▁ ▁Let ▁ ▁be ▁a ▁Se if ert ▁surface ▁for ▁a ▁kn ot , ▁, ▁which ▁can ▁be ▁represented ▁as ▁a ▁disc ▁ ▁with ▁bands ▁attached . ▁The ▁bands ▁will ▁typically ▁be ▁tw isted ▁and ▁kn otted . ▁Each ▁band ▁corresponds ▁to ▁a ▁generator ▁. ▁ ▁can ▁be ▁represented ▁by ▁a ▁circle ▁which ▁travers es ▁one ▁of ▁the ▁bands . ▁Define ▁