Proving that wild cells exist

In their famous paper Fox and Artin constructed several examples of wild cells in 3-space.The present authors construct a wild disk D in the 4-sphere S 4 with the property that the proof of nontameness is perhaps the most elementary possible.We require only the knowledge that if K is the trefoil knot in the 3-sphere S 3 , then the fundamental group π x {S z -K) is not abelian.Parenthetically, the wild disk D constructed here has the property that every arc on D is tame, a fact which follows immediately from the construction. In S3 let {Ki} be a sequence of polygonal trefoil knots that converge to a point q while each K ό lies interior to a 3-simplex that meets no other K { .We consider S 3 as being the equator of S 4 while H is the upper hemisphere of S\ In H-S 3 let {p<} be a sequence of points converging to q.If PiK t is the cone over K t with vertex p if let {Pi} be so chosen that the disks {3><1Q are disjoint in pairs.Now in S 3 join p x K r and p 2 K 2 by a polyhedral disk D x so that pj£ x UAU pj&z is a disk disjoint from ((JΓ ViKi) U q.We next join p z K 2 and p z K z by a polyhedral disk, D 2t in S 3 so that PyK x UAU P%K 2 U A U p 3 K 3 is a disk disjoint from ((JΓ ViKi) U q.This process is continued so that as i -> oo the diameter of Z^ tends to 0 and the disk D is As a subset of S 4 , D is locally tame [1] except perhaps at q. THEOREM.