The Yokonuma–hecke Algebras and the Homflypt Polynomial

Abstract

Abstract. 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.

An isomorphism theorem for Yokonuma–Hecke algebras and applications to link invariants

“…Remarkably, they also produced isotopy invariants for classical links and singular links [13,14]. Even though the obtained invariants for classical links are different from the HOMFLYPT polynomial (excepted in some trivial cases), all the computed examples seem to indicate that the invariants for classical links obtained from Y d,n so far are topologically equivalent to the HOMFLYPT polynomial [2]. In fact, if we restrict to classical knots, such an equivalence has been announced in [1].…”

“…More precisely, among the 2 d − 1 basic Markov traces, there are d of them whose associated invariants coincide with the HOMFLYPT polynomial. These basic Markov traces are the ones indexed by compositions into d parts with only one part equal to 1 and all the others equal to 0 (in the particular case of the Juyumaya-Lambropoulou invariants, this result corrresponds to [2,Corollary 1]). …”

“…Secondly, we will show that this equivalence is true whenever we restrict our attention to classical knots. We refer to [1,2] for similar results about Juyumaya-Lambropoulou invariants. Note that these invariants are shown in Section 6.5 below to be certain linear combinations of the set {Γ .…”

“…The precise connections between the presentation above and the presentation used in [2,10,12,13,14,15] will be carefully investigated in Section 6 (see also [3,Remark 1]). △…”

“…Let q be an indeterminate. In [2,10,12,13,14,15], the Yokonuma-Hecke algebra is presented as a certain quotient of the group algebra over C[q, q −1 ] of the Z/dZ-framed braid group Z/dZ≀B n . Namely, there are generators G 1 , G 2 , .…”