Bracket trace on the three-strand Temperley-Lieb algebra ($\mathrm{tr}(1)={\delta }^2,\mathrm{tr}(e_i)=\delta ,\mathrm{tr}(e_1{e}_2)=1$)

Close a three-strand planar tangle and evaluate its reduced Kauffman bracket. The basis $1,e_1,e_2,e_1{e}_2,e_2{e}_1$ has the displayed trace values, with the last value also valid for $e_2{e}_1$. Writing braid generators as $A1+A^{-1}{e}_i$ turns knot-polynomial computations into a five-dimensional algebra calculation.