▁to ▁be ▁the ▁number ▁of ▁full ▁tw ists ▁in ▁the ▁band ▁mod ulo ▁ 2 . ▁Suppose ▁we ▁let ▁ ▁bound ▁, ▁and ▁push ▁the ▁Se if ert ▁surface ▁ ▁into ▁, ▁so ▁that ▁its ▁boundary ▁still ▁res ides ▁in ▁. ▁Ar ound ▁any ▁generator ▁, ▁we ▁now ▁have ▁a ▁trivial ▁normal ▁ 3 - plane ▁vector ▁bundle . ▁T rivial ise ▁it ▁using ▁the ▁trivial ▁fr aming ▁of ▁the ▁normal ▁bundle ▁to ▁the ▁embedding ▁ ▁for ▁ 2 ▁of ▁the ▁sections ▁required . ▁For ▁the ▁third , ▁choose ▁a ▁section ▁which ▁remains ▁normal ▁to ▁, ▁whilst ▁always ▁remaining ▁tang ent ▁to ▁. ▁This ▁trivial isation ▁again ▁determ ines ▁an ▁element ▁of ▁, ▁which ▁we ▁take ▁to ▁be ▁. ▁Note ▁that ▁this ▁coinc ides ▁with ▁the ▁previous ▁definition ▁of ▁. ▁▁ ▁The ▁Ar f ▁invariant ▁of ▁a ▁kn ot ▁is ▁defined ▁via ▁its ▁Se if ert ▁surface . ▁It ▁is ▁independent ▁of ▁the ▁choice ▁of ▁Se if ert ▁surface ▁( The ▁basic ▁surg ery ▁change ▁of ▁S - equ ival ence , ▁adding / rem oving ▁a ▁t ube , ▁adds / de let es ▁a ▁ ▁direct ▁summ and ), ▁and ▁so ▁is ▁a ▁kn ot ▁invariant . ▁It ▁is ▁add itive ▁under ▁connected ▁sum , ▁and ▁van ishes ▁on ▁slice ▁kn ots , ▁so ▁is ▁a ▁kn ot ▁conc ord ance ▁invariant . ▁▁ ▁The ▁intersection ▁form ▁on ▁the ▁- dimensional ▁- co efficient ▁hom ology ▁ ▁of ▁a ▁fr amed ▁- dimensional ▁manifold ▁M ▁has ▁a ▁quadratic ▁ref in ement