During the work of S.L. and J. Przytycki on the problem of computing the 3rd skein module of the lens spaces L(p, 1) following the above strategy, it turned out that the skein rule of the homfly-pt type invariants in [12, 13, 8] related to t was actually 'artificial', so far that knot invariants in ST were concerned, and Knot theory and B-type Hecke algebras 4 that for analogous constructions in L(p, 1) it was needed to have constructed first the most generic 2-variable Jones analogue in ST ,

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... e that would not satisfy any skein relation involving t. We drop then the quadratic relation of t, and we consider the quotient of the group algebra Z [q ±1 ]B 1,n by factoring out only the relations for all i. This is now a new infinite dimensional algebra, which we denote by H n (q, ∞) and we shall call it generalized Iwahori-Hecke algebra of type B. By g i above we denote the image of σ i in H n (q, ∞), whilst the symbol ∞ was chosen to indicate that the generator t satisfies no order relation (since now any power t k , for k ∈ Z may appear, like in B 1,n ). For connections of these algebras with the affine Hecke algebras of type A see Remark 1.