On the Jones polynomial of closed 3-braids

Abstract

In [J, 2] Vaughan Jones introduced a new polynomial VL(t ) which is an invariant of the isotopy type of an oriented knot or link L c S 3. The polynomial can be computed from an arbitrary representation of L as a closed braid, i.e. from an element in one of the Artin braid groups B,, n = 1, 2, 3 . . . . . It is very powerful, distinguishing the trefoil and its mirror image, the unknot and the Kinoshita-Terasaka knot, and any two members of the infinite sequence of Whitehead links, all of which have homeomorphic complements. In an early version of [J, 2] (before the results reported here were complete) Jones had conjectured that his polynomial was injective on closed 3-braids. In this note we use trace identities in the Burau matrix representation of B 3 to construct myriad counterexamples to that conjecture. At the same time, we also disprove a second conjecture of Jones from [J, 23, that a link L is amphicheiral if VL(t)= VL(t-1). The 2-variable generalized Jones polynomial introduced in [ F Y H L M O ] also fails to distinguish any of our link pairs. Finally, our examples answer in the negative a question of Morton, who asked whether the Alexander polynomial is a complete invariant of the link obtained by adjoining the braid axis to a closed 3-braid link. In the monograph [Mu] Murasugi began a classification of closed 3-braid links, work which was extended by Hartley in [H]. From the partial results in [Mu] and [H] it seemed reasonable to conjecture that, omitting various obvious exceptional cases (links with braid index <3, composite links and torus links) the link type of a closed 3-braid is determined by its conjugacy class in Bj, up to the equivalences which correspond to orientation changes. This conjecture remains open. When we began this work we had hoped to settle it, expecting that the conjugacy class of a braid was determined by the Jones or Alexander polynomial of the associated closed braid. However, this is far from the truth. The problem of characterizing all of the 3-braids whose closures have a given Jones or Alexander or 2-variable polynomial seems to be very non-trivial and possibly even to be too complicate to have an interesting solution.