Classical Link Research Articles

Free coverings and modules of boundary links

Let $L = \{ {K_1}, \ldots ,{K_m}\}$ be a boundary link of $n$-spheres in ${S^{n + 2}}$, where $n \geqslant 3$, and let $X$ be the complement of $L$. Although most of the classical link invariants come from the homology of the universal abelian cover $\tilde X$ of $X$, with increasing $m$ these groups become difficult to manage. For boundary links, there is a canonical free covering ${X_\omega }$, which is simultaneously a cover of $\tilde X$. Thus, knowledge of ${H_ \ast }{X_\omega }$ yields knowledge of ${H_ \ast }\tilde X$. We study general properties of such covers and obtain, for $1 < q < n/2$, a characterization of the groups ${H_q}{X_\omega }$ as modules over the group of covering transformations. Some applications follow.

Concordance implies homotopy for classical links inM 3

Concordance implies homotopy for classical links inM 3

Energy of Formation of Cyclol Molecules

ACCORDING to the cyclol theory of the structure of proteins1—a working hypothesis recently put forward in these columns—the polypeptide, the essential unit in the molecule on the classical theory, is replaced by the cyclized polypeptide (Fig. 1): correspondingly the classical peptide links (joining one pair of C,N atoms) are replaced either by the double peptide links (joining two pairs of C,N atoms) or by the triple peptide link (joining three pairs of C,N atoms) (Fig. 2). The reaction polypeptides r proteins is thus regarded as a ring chain tautomerism2, which takes the form if the appropriate groups are in the imide form, and the form if they are in the amide form.