A new polynomial invariant of knots and links

Abstract

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.

Chern-Simons theory as the large-mass limit of topologically massive Yang-Mills theory

“…In principle, W has local and non-local contributions and, in perturbation theory, can be expressed as the sum 12) with the index i counting the number of fields in W i . Note that we have restricted the sum to i ≥ 2, since contributions to the effective actionΓ at any order in perturbation theory, and in particular toΓ (1) , are at least quadratic in the fields, respectively, we have that…”

Vassiliev invariants for links from Chern-Simons perturbation theory

“…Though one could think that the first approach would lead to simpler expressions, it turns out that this is not the case. The integral expressions forγ 4 3 ,γ 7 4 ,γ 8 4 andγ 9 4 are rather long as compared to the ones obtained in the second approach.…”

“…10. Among all those group factors there are only one primitive group factor at first and second orders,ṡ 11 andṡ 22 , two at order three,ṡ 33 andṡ 34 , and five at order four, s 46 ,ṡ 47 ,ṡ 48 ,ṡ 49 andṡ 4,10 . Using this analysis we can then use (5.7) to write theγ j i in terms of the primitive elements of the associated geometrical factors.…”

“…Apply the formalism described in the previous sections toL. 4. Let ǫ → 0, ǫ ′ → 0 and contract i, j with a δ j i and i ′ , j ′ with a δ j ′ i ′ .…”

Twisting invariance of link polynomials derived from ribbon quasi-Hopf algebras

“…7 and 8. Around the same time as the appearance of quantum algebras was Jones's discovery of a new polynomial invariant, 9 an evaluation of which may be undertaken through the simplest quantum algebra U q (sl(2)) in its minimal ͑two-dimensional͒ representation. After this breakthrough researchers proceeded to obtain generalizations with the notable examples being the HOMFLY 10 and Kauffman 11 invariant polynomials. What soon became apparent was that the series of link polynomials associated with the fundamental representations of the ͑nonexceptional͒ quantum algebras and superalgebras coincided with the two-variable invariants developed in the wake of the discovery of Jones.…”