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The five normal forms in the three-strand algebra

Example

In rank $3$ , the two generators are $u_{1}, u_{2}$ . The admissible Jones normal forms from Jones normal forms span and are counted by Catalan paths are

$1, u_{1}, u_{2}, u_{2} u_{1}, u_{1} u_{2} .$

The two adjacent triple relations reduce alternating length-three words:

$u_{1} u_{2} u_{1} = u_{1}, u_{2} u_{1} u_{2} = u_{2} .$

For example, $u_{2} u_{1} u_{2} u_{1} = (u_{2} u_{1} u_{2}) u_{1} = u_{2} u_{1}$ , while $u_{1} u_{2} u_{1} u_{2} = (u_{1} u_{2} u_{1}) u_{2} = u_{1} u_{2}$ . The five listed words are an $R$ -basis for every commutative coefficient ring and every $δ \in R$ .

Facts & Assumptions

Given: A commutative ring $R$ , $δ \in R$ , and rank $n = 3$ .

[F1]

The parameters of a Jones normal form satisfy $1 \leq k_{i} \leq j_{i} < 3$ , with both $j_{i}$ and $k_{i}$ strictly increasing from block to block. (Jones normal forms span and are counted by Catalan paths)

[F2]

$u_{1}^{2} = δ u_{1}$ , $u_{2}^{2} = δ u_{2}$ , $u_{1} u_{2} u_{1} = u_{1}$ , and $u_{2} u_{1} u_{2} = u_{2}$ . (The abstract Temperley–Lieb generator presentation)

[F3]

The Jones normal forms are linearly independent and span the presented algebra because the presentation map to the diagram algebra is an isomorphism. (The generator presentation is the planar diagram algebra over every commutative ring)

Proof

technique · direct

1.1F1construct

For an empty parameter list the word is $1$ . For one block, the pairs $(j, k) = (1, 1), (2, 1), (2, 2)$ give $u_{1}, u_{2} u_{1}, u_{2}$ . For two blocks, strict increase forces $(j_{1}, j_{2}) = (1, 2)$ and $(k_{1}, k_{2}) = (1, 2)$ , giving $u_{1} u_{2}$ ; three blocks are impossible because only $j = 1, 2$ are available. This lists exactly five forms.

2.1F2F3algebra∎

The reductions in the Example follow by one use of the corresponding triple relation in [F2]; the basis claim follows from [F3]. Since no step divides by $δ$ , these statements include $δ = 0$ and all other ring specializations.

Depends on

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