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The framization is a mechanism designed by Juyumaya and Lambropoulou
and consists of a generalization of a knot algebra via the addition
of framing generators. In this way we obtain a new algebra which is
related to framed knots. More precisely, the framization procedure
can roughly be regarded as the procedure of adding framing
generators to the generating set of a knot algebra, of defining
interacting relations between the framing generators and the
original generators of the algebra and of applying framing on the
original defining relations of the algebra. The resulting framed
relations should be topo-logically consistent. The most difficult
problem in this procedure is to apply the framization on the
relations of polynomial type.
In this talk we will present three framizations of the
Temperley--Lieb algebra as a quotient of the Yokonuma--Hecke algebra
over appropriate two-sided ideals. The quotient algebras that arise
are: the Framization of the Temperley-Lieb algebra
\(\text{FTL}_{d,n}(u)\), the Yokonuma-Temperley-Lieb algebra
\(\text{YTL}_{d,n}(u)\) and the Complex Reflection Temperley-Lieb
algebra \(\text{CTL}_{d,n}(u)\). From these we choose the algebra
\(\text{FTL}_{d,n}(u)\) as the analogue of the Temperley-Lieb algebra
in the context of framing, since it reflects the construction of a
"framed Jones Polynomial" in the most natural way.
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