Narrow Results

This text is a brief description of the Jones polynomial as it appears in the announcement in [9], together with Jones’ comments in the letter he addressed to Joan Birman on 31st May 1984 [8]. As much as possible I have followed the author’s initial notations. I add comments about the history of the events as I remember them, with the help of several testimonies of the “actors”. Among them: Didier Hatt-Arnold, Pierre de la Harpe, Cam Van Quach Hongler. It took me nontrivial efforts with my memories to get the chronology right (I hope so). The help of Eva Bayer was decisive. My warmest thanks to her. One may disagree with some of my statements which express a personal opinion. On purpose I have adopted a candid style.

Personal recollections of meetings with Vaughan Jones.

We show that there is a unique Markov trace on the tower of Temperley–Lieb type quotients of Hecke algebras of Coxeter type E_n (for all n ≥ 6). We explain in detail how this trace may be computed easily using tom Dieck's calculus of diagrams. As applications, we show how to use the trace to show that the diagram representation is faithful, and to compute leading coefficients of certain Kazhdan–Lusztig polynomials.

In this paper we introduce a Jones-type invariant for singular knots, using a Markov trace on the Yokonuma–Hecke algebras Y_d,n(u) and the theory of singular braids. The Yokonuma–Hecke algebras have a natural topological interpretation in the context of framed knots. Yet, we show that there is a homomorphism of the singular braid monoid SB_n into the algebra Y_d,n(u). Surprisingly, the trace does not normalize directly to yield a singular link invariant, so a condition must be imposed on the trace variables. Assuming this condition, the invariant satisfies a skein relation involving singular crossings, which arises from a quadratic relation in the algebra Y_d,n(u).

We develop a diagrammatic formalism for calculating the Alexander polynomial of the closure of a braid as a state-sum. Our main tools are the Markov trace formulas for the HOMFLY-PT polynomial and Young's semi-normal representations of the Iwahori–Hecke algebras of type A.

Using the Fourier expansion of Markov traces for Ariki–Koike algebras over ℚ(q, u₁, …, u_e), we give a direct definition of the Alexander polynomials for mixed links. We observe that under the corresponding specialization of a Markov parameter, the Fourier coefficients of Markov traces take quite a simple form. As a consequence, we show that the Alexander polynomial of a mixed link is essentially equal to the Alexander polynomial of the link obtained by resolving the twisted parts.

Kashaev and Reshetikhin proposed a generalization of the Reshetikhin–Turaev link invariant construction to tangles with a flat connection in a principal G-bundle of the complement of the tangle. The purpose of this paper is to adapt and renormalize their construction to define invariants of G-links using the semi-cyclic representations of the non-restricted quantum group associated to 𝔰𝔩(2), defined by De Concini and Kac. Our construction uses a modified Markov trace. In our main example, the semi-cyclic invariants are a natural extension of the generalized Alexander polynomial invariants defined by Akutsu, Deguchi and Ohtsuki. Surprisingly, direct computations suggest that these invariants are actually equal.

We compare the invariants for classical knots and links defined using the Juyumaya trace on the Yokonuma–Hecke algebras with the HOMFLYPT polynomial. We show that these invariants do not coincide with the HOMFLYPT except in a few trivial cases.

We define a tower of affine Temperley–Lieb algebras of type Ã. We prove that there exists a unique Markov trace on this tower, this trace comes from the Markov–Ocneanu–Jones trace on the tower of Temperley–Lieb algebras of type A. We define an invariant of special kind of links as an application of this trace.

In this paper, we study properties of the Markov trace tr${}_d$ and the specialized trace ${\mathrm{\text{tr}}}_{d,D}$ on the Yokonuma–Hecke algebras, such as behavior under inversion of a word, connected sums and mirror imaging. We then define invariants for framed, classical and singular links through the trace ${\mathrm{\text{tr}}}_{d,D}$ and also invariants for transverse links through the trace tr${}_d$. In order to compare the invariants for classical links with the Homflypt polynomial, we develop computer programs and we evaluate them on several Homflypt-equivalent pairs of knots and links. Our computations lead to the result that these invariants are topologically equivalent to the Homflypt polynomial on knots. However, they do not demonstrate the same behavior on links.

We describe an algorithm for computing the invariants of classical links arising from the Yokonuma–Hecke algebras. A detailed overview of the algorithm is given, following closely its implementation, a program used to calculate the invariants on several Homflypt-equivalent pairs of links.

We introduce the concept of tied links in the solid torus, which generalizes naturally the concept of tied links in $S^3$ previously introduced by Aicardi and Juyumaya. We also define an invariant of these tied links by using skein relations, and we then recover this invariant by using Jones’ method over the bt-algebra of type $B$ and the Markov trace defined on this.

We construct a polynomial link invariant as Markov trace on certain one parameter algebras and we prove that it is equal to the invariant corresponding to the exeptional Lie algebra of type G₂. We use braid representatives to calculate the invariant for several knots and links.

The notion of Vassiliev algebra in case of handlebodies is developed. Analogues of the results of J. Baez for links in handlebodies are proved. This implies that there is a one-to-one correspondence between the special class of finite type invariants of links in hanlebodies and the homogeneous Markov traces on Vassiliev algebras. This approach uses the singular braid monoid and braid group in a handlebody and the generalizations of the theorem of J. Alexander and the theorem of A. A. Markov for singular links and braids and the relative version of Markov's theorem.