The factorization of simple knots

Abstract

Abstract. For high-dimensional simple knots we give two theorems concerning unique factorization into irreducible knots, and provide examples to show that the hypotheses are necessary in each case. IntroductionThe purpose of this paper is to collate and extend the known results on the factorization of high dimensional knots. By an n-knot we mean an oriented smooth or locally flat PL pair (S n+2 , S m ), where S n is homeomorphic to the n-sphere S n . The sum Jc + l of two n-knots k and I is obtained by excising the interior of a tubular neighbourhood of a point on each S n and identifying the boundaries of the resulting knotted ball pairs so that the orientations match up. A knot k is irreducible if it cannot be written as the sum of two non-trivial knots. It is a result of H. Schubert (16) that for n = I, every knot factorizes into finitely many irreducibles, and that factorization is unique (up to the order of the factors).Given an n-knot k, the exterior K is the closed complement of a tubular neighbourhood of 2™. The knot k is simple if K has the homotopy [(n -l)/2]-type of a circle; that is TJ^K) ~ n 1 (S 1 ) for 1 ^ i < (n-l)/2. For n > 3, this is the most that can be asked without making k trivial (see (ll, 12)). The knot k is fibred if K is fibred over the circle, and we let R denote the infinite cyclic cover of K.In Section 1 we give a short proof that every simple n-knot, n ^ 3, factorizes into finitely many irreducibles. A more general result was published by A. B. Sosinskii in (18), but note the assertion of T. Maeda in (14).Let k be a simple (2q-l)-knot, q ^ 2. There are two ways of classifying such knots in terms of algebraic invariants. The first of these, due to J. Levine, is in terms of the S-equivalence class of the Seifert matrix of k; details may be found in (12). The second method uses the Blanchfield duality pairing, <,>:A o is the field of fractions of A, and H q (K) is regarded as a A-module. Details of this method may be found in (7,8,20,21).Each such knot k has associated with it a quadratic form, as outlined in Section 2. If this form is definite, then k is said to be definite. The knot k is fibred if and only if the leading coefficient of its Alexander polynomial is + 1; this follows easily from the results of R.H.Crowell(3) and W.Browder and J. Levine(2). In Section 2 we show

New Strategies for Identifying Natural Products of Ecological Significance from Corals

“…Within each phylum above the majority of substances belong to a single class of sponges (Porifera: Demospongiae) and one class of corals (Cnidaria: Anthozoa; Fig. 1) [5,[13][14][15]. The majority of coral compounds are derived from the isoprenoid Studies in Natural Products Chemistry Others (1900) Subclass: Hexacorallia = Zoantharia (154) Subclass: Octocorallia (1324) biosynthetic pathway (82%) and the remainder split between acetogenin and amino acid (Figs.…”

“…The pattern of this vibronic structure is given by the Franck-Condon factors [102], which are integrals involving vibrational wave functions of ground and excited electronic states. From the vibrational level v i ¼ 0 of the ground electronic level, the intensity of the transitions to excited state vibrational levels is proportional to the overlap integral value, hf |ti, between the correspondent functions: (13) where |ii and hf | are the bra-ket notations for the initial vibrational wave functions, C i , and the final (conjugated complex), C f *. The intensity of an absorption band in function of the radiation frequency, I A (o), is given by [3][4][5] …”

Mutation of knots

“…Thus we have an example in which the knot module of k + k is not isomorphic to that of k + k'. Another example can be obtained from [BHK,§4], and other examples can be found using [B] and number theory tables.…”

Simple Non-Finite Knots Are Not Prime in Higher Dimensions

“…And k is irreducible if k = a + b implies that either a or b is the trivial knot. It is well known that factorisation is not unique for knots in higher dimensions, so that irreducible does not imply prime; see, for example, [1,2,3,4,10]. The object of this paper is to show that, in higher dimensions, simple non-finite knots are never prime.…”