A note on spun knots

Abstract

This paper contains some remarks about spinning which show, in particular, that the spun reef knot is equivalent to the spun granny. I am grateful to J. J. Andrews for having some time ago drawn my attention to the following problem concerning spun knots: Is the spun reef equivalent to the spun granny? Since it seems that several knot theorists have at one time or another been interested in this problem, it may be worth recording the following simple solution. Spinning was introduced by Artin [2], and the more general process of p-spinning seems to have been used first by Epstein [4]. Since then it has been studied and used by many other authors; see [1], [3], [5], [7], etc. The definition (which is valid in DIFF or PL) is probably best expressed as follows. Let K = (M, N) be a pair consisting of a closed, oriented, manifold M of dimension m, and a closed, oriented (locally flat) submanifold N of dimension n. Let K = (M, N) be the pair obtained by removing the interior of a standard ball pair (B m, BI) c (M, N). Then the p-spin of K, p > 0, is the oriented pair ap(K) =a (K x DP+'), where DP`1 is the (p + 1)-disc with some fixed orientation. In particular, if K is a knot of S' in s', then ap(K) is a knot of S`P in Let denote orientation-preserving isomorphism (of pairs). It is well known that p-spinning behaves nicely with respect to connected sum: LEMMA 1. up (K1 # K2)up(K) ? ap(K2). PROOF. Using # to denote boundary connected sum, we have gp(KI # K2) =a((K, # K2) x DP+1) K2) x DP+') -a((K' x DP+') (K2 x DP+D) a VI x DP+') # a(K2 X DP+') = up(KI) # up(K2). Received by the editors December 19, 1974. AMS (MOS) subject classifications (1970). Primary 55A25, 57C45.