Definite hermitian forms and the cancellation of simple knots

Abstract

Schubert has shown that every classical knot 2:1 c S 3 factorises uniquely into the connected sum of finitely many indecomposable knots (cf. [12]). In particular cancellation holds for these knots. For higher elimensional simple knots factorisation is not always unique (cs [5] and [1]), but in many cases we still have cancellation (see [2], Proposition 6.6).In this note we shall give counter examples to the cancellation of non-singular hermitian and skew-hermitian forms. In order to obtain these examples we shall show that the extension of the 2J-lattice/~4~, n ~= 1, to certain orders is indecomposable.Using the classification of simple (2q--1)-knots 2:2q-lc S2q +1, q ~= 1, in terms of (--1)q+l-hermitian (Blanchfield) forms, we shall then prove that cancellation does not hold for higher odd-dimensional knots.I thank Hans-Joehen Bartels and Larry Gerstein for useful conversations.

Cancellation of hyperbolic ε-hermitian forms and of simple knots

“…For n odd, counter-examples to the cancellation of n-knots have been given in [2]. In the present note we shall show that the cancellation problem also has a negative solution for n even, n ^ 4.…”

“…Therefore K x # K is isotopic to K 2 # K, but K 1 is not isotopic to K 2 . Notice that these knots are not fibred, whereas the counter-examples of [2] are fibred.…”