On the additivity of the clasp singularities
Kanji Morimoto · Institutional Repositories DataBase (IRDB) · 1987
For a tame knot K in S3, several numerical invariants are defined, and relations between those invariants and a connected sum of knots have been studied by several authors ([1], [5], [6], [7], [8], [9], [10]).For example, for g(K), the genus of K, g(K 1 lfK 2 )=g(K 1 )+g(K 2 ) holds, where K 1 lfK 2 denotes a connected sum of K 1 and K 2 (cf.[5], [7]).And for u(K), the unknotting number of K, it was proved by M. G. Scharlemann in [6] that if u(K) = 1 then K is a prime knot.Now let c(K) be the clasp number of K, the definition of c(K) will be given later.Then the following equality is unknown.In this paper we will prove the following theorem.THEOREM.Let K 1 and K 2 be two non-trivial knots.If c(K 1 lfK 2 )=2, Then, together with Proposition 2, we get the following corollary.Throughout this paper we will work in the piecewise linear category.Cl(.), Int ( •) and o( •) mean the closure, the interior and the boundary respectively.For a set A, IAI denotes the cardinal number of A.