The purpose of this note is to announce a new isotopy invariant of oriented links of tamely embedded circles in 3-space.We represent links by plane projections, using the customary conventions that the image of the link is a union of transversely intersecting immersed curves, each provided with an orientation, and undercrossings are indicated by broken lines. Following Conway [6], we use the symbols L+, Lo, L_ to denote links having plane projections which agree except in a small disk, and inside that disk are represented by the pictures of Figure 1.Conway showed that the one-variable Alexander polynomials of L+, Lo, L_ (when suitably normalized) satisfy the relation

The purpose of this note is to announce a new isotopy invariant of oriented links of tamely embedded circles in 3-space.We represent links by plane projections, using the customary conventions that the image of the link is a union of transversely intersecting immersed curves, each provided with an orientation, and undercrossings are indicated by broken lines. Following Conway [6], we use the symbols L+, Lo, L_ to denote links having plane projections which agree except in a small disk, and inside that disk are represented by the pictures of Figure 1.Conway showed that the one-variable Alexander polynomials of L+, Lo, L_ (when suitably normalized) satisfy the relation It was evident from the circumstances that the four groups arrived at their results completely independently of each other, although all were inspired by the work of Jones (cf. [10], and also [8, 9]). The degree of simultaneity was such that, by common consent, it was unproductive to try to assess priority. Indeed it would seem that there is enough credit for all to share in.Each of these papers was refereed, and we would have happily published any one of them, had it been the only one under consideration. Because the alternatives of publication of all four or of none were both unsatisfying, all have agreed to the compromise embodied here of a paper carrying all six names as coauthors, consisting of an introductory section describing the basics written by a disinterested party, and followed by four sections, one written by each of the four groups, briefly describing the highlights of their own approach and elaboration.

No abstract

The fundamental group of a 2-bridge knot has a particularly nice presentation, having only two generators and a single relation. For certain families of 2-bridge knots, such as the torus knots, or the twist knots, the relation takes on an especially simple form. Exploiting this form, we derive a formula for the A-polynomial of twist knots. Our methods extend to at least one other infinite family of (non-torus) 2-bridge knots. Using these formulae we determine the associated Newton polygons. We further prove that the A-polynomials of twist knots are irreducible.

We extend the Jones polynomial for links in S3 to links in L(p, q), p>0. Specifically, we show that the (2, ∞)-skein module of L(p, q) is free with [p/2]+1 generators. In the case of S1×S2 the skein module is infinitely generated.