Narrow Results

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 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.

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.

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.