The parallel version of polynomial invariants of links
Given a link X, the (companion) link \(X^{(r)}\), called the r-parallel version of X, may be obtained, roughly speaking, from a regular projection of X by drawing r-1 parallels to X, and then untwisting them once at each crossing so the r parallel links have pairwise linking numbers zero. It is shown that equivalent links have equivalent r- parallel versions. The force of this result is that it enables the definition of very strong invariants. For example, knots which by \textit{H. R. Morton} and \textit{H. B. Short} [Math. Proc. Camb. Philos. Soc. 101, 267-278 (1986; Zbl 0629.57001)] and \textit{S. Yamada} [On the 2-variable Jones polynomial of satellite links, in: Topology and Computer Science, 295-300, Kinokuniya, Tokyo (1987)], are indistinguishable by their Jones polynomial have distinct Jones polynomials for their 2-parallel versions. Let \(b^\wedge\) denote the closure of an n-braid b. Let \(B_ n\) denote the braid group on n strands. A link invariant \(\phi\) is said to be of trace type if \(\phi(b^\wedge)\) can be written as a linear combination of characters of representations of \(B_ n\), and these characters satisfy some compatibility conditions with respect to the inclusion of \(B_ n\) in \(B_{n+1}\). (The 2-variable Jones polynomial and the Kauffman polynomial are both of trace type.) This paper gives an efficient method for calculating trace type invariants of \(X^{(r)}\), thus invariants of X. A main result of the paper relates the Jones polynomial of \(b^\wedge\) and the traces of certain representations of b, where b is a 3-braid whose closure is \(b^\wedge\) and \(b^\wedge\) is a knot. Other results concern mutant knots and cable links.
scientific article; zbMATH DE number 4155454