Polynomial invariants of links satisfying cubic skein relations

Abstract

The aim of this paper is to define two link invariants satisfying cubic skein relations. In the hierarchy of polynomial invariants determined by explicit skein relations they are the next level of complexity after Jones, HOMFLY, Kauffman and Kuperberg's G 2 quantum invariants. Our method consists of the study of Markov traces on a suitable tower of quotients of cubic Hecke algebras extending Jones approach.

The BMR freeness conjecture for the tetrahedral and octahedral families

“…Deformed Coxeter group algebras. Let W be an exceptional group of rank 2 and let C be a finite Coxeter group of type either A 3 , B 3 or H 3 with Coxeter system y 1 , y 2 , y 3 and Coxeter matrix (m ij ). We setZ := Z e 2πi m ij .…”

“…• HG 5 := s, t | stst = tsts, (z k u1u2 + z k t −1 su2), where z = (st) 2 and u1 and u2 denote the subalgebras of HG 5 generated by s and t, respectively. 3 and u2 denotes the subalgebra of HG 6 generated by t.…”

“…C3. k ∈ {0, 1}: We expand u 3 as R + Ru + Ru 2 + Ru 3 and by proposition B.42 (ii), (iii) and (v) we have that z k tu 3…”

Markov Traces on the Funar Algebra

“…Let k be a commutative ring with 1 = 0. Given α, β ∈ k, we define the k-algebra K n = K n (α, β) = K n (α, β; k) as the quotient of the group algebra kB n by the relations σ 3 1 − ασ 2 1 + βσ 1 − 1 = 0 (1) and yxy =2α − β 2 − (x + y) − (α 2 − β)(x +ȳ) + β(xy + yx) + α(xȳ + yx +xy +ȳx)…”

“…Up to a change of the sign of β (for the sake of symmetricity), our definition of K n is equivalent to the definition given by Bellingeri and Funar in [1]. Our relation (2) is much shorter than the corresponding relation in [1] (see [1; (2) and Table 1]) because we use σ −1 i instead of σ 2 i . Multiplying (2) by σ 1 from the left or from the right, and simplifying the result using (1) and the braid group relations, we obtain yxȳ =2β − α 2 − (x +ȳ) − (β 2 − α)(x + y) + α(xȳ +ȳx) + β(xy +ȳx + xȳ + yx)…”