Ruth Lawrence

Homological representations of the Hecke algebra

In this paper a topological construction of representations of theAn(1)-series of Hecke algebras, associated with 2-row Young diagrams will be given. This construction gives the representations in terms of the monodromy representation obtained from a vector bundle on which there is a natural flat connection. The fibres of the vector bundle are homology spaces of configuration spaces of points in C, with a suitable twisted local coefficient system. It is also shown that there is a close correspondence between this construction and the work of Tsuchiya and Kanie, who constructed Hecke algebra representations from the monodromy ofn-point functions in a conformal field theory onP1. This work has significance in relation to the one-variable Jones polynomial, which can be expressed in terms of characters of the Iwahori-Hecke algebras associated with 2-row Young diagrams; it gives rise to a topological description of the Jones polynomial, which will be discussed elsewhere [L2].

Asymptotic Expansions of Witten-Reshetikhin-Turaev Invariants for Some Simple

For any Lie algebra g and integral level k, there is defined an invariant Zk*(M, L) of embeddings of links L in 3‐manifolds M, known as the Witten–Reshetikhin–Turaev invariant. It is known that for links in S3, Zk*(S3, L) is a polynomial in q=exp (2πi/(k+cgv), namely, the generalized Jones polynomial of the link L. This paper investigates the invariant Zr−2*(M,○/) when g =sl2 for a simple family of rational homology 3‐spheres, obtained by integer surgery around (2, n)‐type torus knots. In particular, we find a closed formula for a formal power series Z∞(M)∈Q[[h]] in h=q−1 from which Zr−2*(M,○/) may be derived for all sufficiently large primes r. We show that this formal power series may be viewed as the asymptotic expansion, around q=1, of a multivalued holomorphic function of q with 1 contained on the boundary of its domain of definition. For these particular manifolds, most of which are not Z‐homology spheres, this extends work of Ohtsuki and Murakami in which the existence of power series with rational...

3

We write a formula for the LMO invariant of a rational homology sphere presented as a rational surgery on a link inS3. Our main tool is a careful use of the Århus integral and the (now proven) “Wheels” and “Wheeling” conjectures of B-N, Garoufalidis, Rozansky and Thurston. As steps, side benefits and asides we give explicit formulas for the values of the Kontsevich integral on the Hopf link and on Hopf chains, and for the LMO invariant of lens spaces and Seifert fibered spaces. We find that the LMO invariant does not separate lens spaces, is far from separating general Seifert fibered spaces, but does separate Seifert fibered spaces which are integral homology spheres.

-Manifolds

Abstract In this paper the new concept of an n -algebra is introduced, which embodies the combinatorial properties of an n -tensor, in an analogous manner to the way ordinary algebras embody the properties of compositions of maps. The work of Turaev and Viro on 3-manifold invariants is seen to fit naturally into the context of 3-algebras. A new higher dimensional version of Yang-Baxters equation, distinct from Zamolodchikovs equation, which resides naturally in these structures, is proposed. A higher dimensional analogue of the relationship between the Yang-Baxter equation and braid groups is then seen to exhibit a similar relationship with Manin and Schechtmans higher braid groups.

A RATIONAL SURGERY FORMULA FOR THE LMO INVARIANT

A brief review is given of some mathematical aspects of human colour perception and of a standard method for computing the colour of a dyed material. This paper describes an extension of this method to incorporate fluorescent transfer of light between wavelengths and gives an algorithm for the analysis of the discrete case in which this transfer occurs at a finite number of wavelengths. Then, a continuum model is formulated in which transfer may occur from any wavelength to any longer wavelength, and this leads to the question of determining the coefficients in an integrodifferential equation from knowledge of its solutions; it is solved by an algorithm derived from that in the discrete case.

Algebras and triangle relations

By analysing Ohtsukis original work in which he produced a formal power series invariant of rational homology 3-spheres, we obtain a simplified explicit formula for them, which may also be compared with Rozanskys integral expression. We further show their relation to the exact SO(3) Witten-Reshetikhin-Turaev invariants at roots of unity in a stronger form than that given in Ohtsukis original work.

Fluorescent transfer of light in dyed materials

Abstract Using the R -matrix formulation of the sl 3 invariant of links, we compute the coloured sl 3 generalised Jones polynomial for the trefoil. From this, the PSU (3) invariant of the Poincare homology sphere is obtained. This takes complex number values at roots of unity. The result obtained is formally an infinite sum, independent of the order of the root of unity, which at roots of unity reduces to a finite sum. This form enables the derivation of the PSU (3) analogue of the Ohtsuki series for the Poincare homology sphere, which it was shown by Thang Le could be extracted from the PSU (N) invariants of any rational homology sphere.

ON OHTSUKI'S INVARIANTS OF HOMOLOGY 3-SPHERES.

This paper addresses the problem of constructing higher dimensional versions of the Yang?Baxter equation from a purely combinatorial perspective. The usual Yang?Baxter equation may be viewed as the commutativity constraint on the two-dimensional faces of a permutahedron, a polyhedron which is related to the extension poset of a certain arrangement of hyperplanes and whose vertices are in 1?1 correspondence with maximal chains in the Boolean poset Bn. In this paper, similar constructions are performed in one dimension higher, the associated algebraic relations replacing the Yang?Baxter equation being similar to the permutahedron equation. The geometric structure of the poset of maximal chains inSa1×?×Sakis discussed in some detail, and cell types are found to be classified by a poset of “partitions of partitions” in much the same way as those for permutahedra are classified by ordinary partitions.

The PSU(3) invariant of the Poincaré homology sphere

We give a self-contained treatment of Le and Habiros approach to the Jones function of a knot and Habiros cyclotomic form of the Ohtsuki invariant for manifolds obtained by surgery around a knot. On the way we reproduce a state sum formula of Garoufalidis and Le for the colored Jones function of a knot. As a corollary, we obtain bounds on the growth of coefficients in the Ohtsuki series for manifolds obtained by surgery around a knot, which support the slope conjecture of Jacoby and the first author.

Yang-Baxter Type Equations and Posets of Maximal Chains

It has been seen elsewhere how elementary topology may be used to construct representations of the Iwahori-Hecke algebra associated with two-row Young diagrams, and how these constructions are related to the production of the same representations from the monodromy of n-point correlation functions in the work of Tsuchiya & Kanie and to the construction of the one-variable Jones polynomial. This paper investigates the extension of these results to representations associated with arbitrary multi-row Young diagrams and a functorial description of the two-variable Jones polynomial of links in S3.