Temperley and Lieb presented a transfer‐matrix approach to study problems related to the study
of percolation and coloring problems on an infinite 2‐dimensional lattice. They noted that the
transfer matrices respected a certain list of relations, which led to the introduction and study of the
so‐called Temperley‐‐Lieb algebras. Temperley‐‐Lieb algebras are intimately related to knot theory.
The Jones polynomial of a knot can be derived from a representation of the braid group into the
Temperley‐Lieb algebra, the so‐called Burau representation. The "Jones unknotting conjecture"
states that Jones polynomials distinguish the unknot from nontrivial knots. In this talk, we will
introduce Jones polynomials through Temperley‐Lieb algebras and describe a current program that
aims to find a nontrivial knot with a trivial Jones polynomial through Burau's unfaithful
representation.