Some noninvertible links

Abstract

Abstract. We construct for each n ≥ 2 infinitely many n-component oriented links that are neither amphicheiral nor invertible; among these examples, infinitely many are Brunnian links.Amphicheirality and invertibility questions concerning knots and links in the 3-sphere S 3 are among the earliest problems in classical knot theory. These questions are of interest since they add to our knowledge of knot and link groups. The Jones polynomial and its various generalizations detect nonamphicheirality of a wide range of knots and links, but they fail to distinguish a single knot or link from its inverse.In this paper, we construct for each n ≥ 2 infinitely many n-component oriented links that are neither amphicheiral nor invertible; among these examples, infinitely many are Brunnian links. Our starting observation is that if an oriented link L in S It should be noticed that the existence of noninvertible knots was first established by Trotter [7]. Noninvertible links with an arbitrary number of components and invertible proper sublinks were first constructed by Whitten [8], [9]. However, Whitten's examples are not Brunnian links and the noninvertibility was established by difficult arguments from combinatorial group theory. Milnor'sμ-invariants [4], [5] of weight m have the ability to detect nonamphicheirality or noninvertibility of oriented links for m even or odd respectively. These invariants can be used to show that the n-component Brunnian link in Figure 7 of [4] is nonamphicheiral or noninvertible for n even or odd respectively. By using the algorithm provided by Cochran [1], further nonamphicheiral or noninvertible Brunnian links could be constructed. However, Milnor'sμ-invariants fail to detect noninvertibility of 2-component oriented links by Traldi [6] and it is not clear whether they detect noninvertibility of n-component oriented links for other even n.Definitions. An n-component oriented link in an oriented 3-manifold X is an ordered collection of n disjoint oriented circles tamely embedded in X. Two ncomponent oriented links L 1 and L 2 are equivalent or have the same link type if there exists an orientation preserving homeomorphism h of X onto itself such that

Hyperbolic Brunnian links

“…Then we are interested in how many Brunnian links in history, which, to our best knowledge, all appear in [6,8,20,1,3,2,15,12,10,18,16,5], are hyperbolic. Our methods work to detect s-primeness for all of them and untiedness for all except Fountains and Jade-pendants in [5].…”

New criteria and constructions of Brunnian links

“…The first examples were introduced by H. Brunn [8] in 1892 but it was not until 1961 their Brunnian property was proved by DeBrunner [9]. Since then, various authors constructed many kinds of examples, which, as far as we know, all appear in [20,5,6,4,15,12,10,16,17,13]. Recently, Nils A. Baas et al [1,2,3,4,6] constructed several infinite families of Brunnian links, vastly expanding our knowledge in this area.…”