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The Temperley-Lieb quotient and the Jones specialization

Definition

Assume the Axiom of Choice for the link-invariance assertion below, via The Hecke trace construction is an oriented link invariant and its Markov-equivalence supplier. Let $H (1) \subset H (2) \subset \dots$ be the Hecke tower over $Λ = Z [v^{\pm 1}, z]$ of The Markov trace on the type-A Hecke tower, with $H (n)$ generated by $T_{1}, \dots, T_{n - 1}$ (The generic type-A Hecke algebra). For $1 \leq i \leq n - 2$ let $E_{n}^{(i)} : = \sum_{w \in ⟨ s_{i}, s_{i + 1} ⟩} T_{w}$ be the sum of the six standard-basis elements over the parabolic subgroup $⟨ s_{i}, s_{i + 1} ⟩ ≅ S_{3}$ inside $H (n)$ , and let $J (n) \subset H (n)$ be the two-sided ideal generated by the elements $E_{n}^{(i)}$ for $1 \leq i \leq n - 2$ (Left, right and two-sided ideals). The Temperley--Lieb quotient of the Hecke tower is the quotient algebra $T L (n) : = H (n) / J (n)$ (The quotient ring $R / I$ with $(r + I) (s + I) = r s + I$ ). Put $D : = Λ [(v + 1)^{- 1}]$ and form $H_{D} (n) : = D \otimes_{Λ} H (n)$ and ${T L}_{D} (n) : = D \otimes_{Λ} T L (n)$ . The normalized generators below are defined in these scalar extensions, where $v + 1$ is a unit.

The Temperley--Lieb relations. In $H_{D} (n)$ put $e_{i} : = \frac{T_{i} + 1}{v + 1} (1 \leq i \leq n - 1), λ : = \frac{v}{(v + 1)^{2}} .$ Then $e_{i}^{2} = e_{i}$ and $e_{i} e_{j} = e_{j} e_{i}$ for $∣ i - j ∣ > 1$ , while in the quotient ${T L}_{D} (n)$ one has $e_{i} e_{i + 1} e_{i} = λ e_{i}, e_{i + 1} e_{i} e_{i + 1} = λ e_{i + 1} (1 \leq i \leq n - 2) .$ These are Jones' Temperley--Lieb relations $e_{i}^{2} = e_{i}$ , $e_{i} e_{i \pm 1} e_{i} = λ e_{i}$ , $e_{i} e_{j} = e_{j} e_{i}$ for $∣ i - j ∣ > 1$ with loop parameter $λ$ . After adjoining $s$ with $s^{2} = v$ to $D$ , the elements $f_{i} : = (v + 1) s^{- 1} e_{i} = λ^{- 1 / 2} e_{i}$ satisfy $f_{i}^{2} = δ f_{i}, f_{i} f_{i + 1} f_{i} = f_{i}, f_{i + 1} f_{i} f_{i + 1} = f_{i + 1}, f_{i} f_{j} = f_{j} f_{i} (∣ i - j ∣ > 1),$ with $δ : = λ^{- 1 / 2} = \frac{v + 1}{v^{1 / 2}} = s + s^{- 1}$ when $s^{2} = v$ .

The Jones specialization. Let $z_{0} : = - \frac{1}{v + 1}$ and let $T : = Z [v^{\pm 1}, (v + 1)^{- 1}, s] / (s^{2} - v)$ be the ring in which $v + 1$ is inverted and a square root $s$ of $v$ is adjoined. By the universal property of the coefficient ring $R$ of The HOMFLYPT coefficient ring there is a unique unital ring homomorphism $φ : R \to T$ with $v \mapsto v, z \mapsto z_{0}, s \mapsto s, u \mapsto s;$ it sends $α = (u z)^{- 1} \mapsto - \frac{s^{2} + 1}{s}$ . The Jones specialization of the invariant $P$ of The HOMFLYPT polynomial from the Hecke Markov trace is $V (\hat{β}) : = φ (P (\hat{β})) = s^{e (β)} (- \frac{s^{2} + 1}{s})^{n - 1} {tr}_{n} (π_{n} (β)) ∣_{z = z_{0}} \in T$ for $β \in B_{n}$ with $n \geq 1$ and closure $\hat{β}$ (The closure of a geometric braid). Then $V$ is an invariant of nonempty oriented links with $V (unknot) = 1$ and $s^{- 2} V (L_{+}) - s^{2} V (L_{-}) = (s - s^{- 1}) V (L_{0})$ on every skein triple; in the variable $t : = s^{2}$ this is the Jones skein relation $t^{- 1} V_{+} - t V_{-} = (t^{1 / 2} - t^{- 1 / 2}) V_{0}$ , and $V$ is the Jones polynomial in the convention of Birman--Brendle §4.3 property 6: positive $σ_{1}^{3}$ has value $t + t^{3} - t^{4}$ . Mirroring a link replaces $t$ by $t^{- 1}$ ; the convention is fixed, rather than chosen separately for each computation. The separate empty-link extension of $P$ specializes to $V (\emptyset) = - s / (s^{2} + 1)$ in $T$ ; that formal extension is outside the classical Jones-polynomial identification for nonempty links.

Caveats. The quotient $T L (n)$ is defined over $Λ$ , but the normalized generators and their displayed relations are in the base change ${T L}_{D} (n)$ , since $v + 1$ need not be a unit in $Λ$ . No claim is made here that the Ocneanu trace on $H (n)$ factors through the quotient map $H (n) \to T L (n)$ ; the quotient is recorded for the Temperley--Lieb relations (1), and the Jones invariant is defined as the specialization of the link invariant $P$ , not as a trace on the quotient. What is proved below is the Markov normalization after scalar extension, namely ${tr}_{n + 1}^{+} (x e_{n}) = \frac{z + 1}{v + 1} {tr}_{n}^{+} (x)$ for $x \in H_{D} (n)$ , whose value at $z_{0}$ is $λ$ ; this is Jones' Markov trace normalization for the Temperley--Lieb parameter $τ = λ$ . The identification of $V$ with the Jones polynomial uses the explicit Hecke-trace specialization in Birman--Brendle §4.3 property 6, quoted in [F6]. That source also supplies the relation on arbitrary oriented skein triples; the local skein supplier proves the braided triples.

Facts & Assumptions

Given: AC (The Axiom of Choice), the Hecke tower over $Λ = Z [v^{\pm 1}, z]$ , its scalar extension to $D = Λ [(v + 1)^{- 1}]$ , the elements $e_{i}$ , the ideals $J (n)$ , the quotients $T L (n)$ and ${T L}_{D} (n)$ , the coefficient ring $R$ and its specialization $φ : R \to T$ .

[F1]

$H (n)$ is the $Λ$ -algebra with generators $T_{1}, \dots, T_{n - 1}$ and relations $T_{i}^{2} = (v - 1) T_{i} + v$ , $T_{i} T_{i + 1} T_{i} = T_{i + 1} T_{i} T_{i + 1}$ and $T_{i} T_{j} = T_{j} T_{i}$ for $∣ i - j ∣ > 1$ , with standard basis ${T_{w} : w \in S_{n}}$ (The generic type-A Hecke algebra, The Markov trace on the type-A Hecke tower).

[F2]

A quotient ring is the universal ring in which the ideal is killed, and a two-sided ideal is closed under left and right multiplication (The quotient ring $R / I$ with $(r + I) (s + I) = r s + I$ , Left, right and two-sided ideals).

[F3]

The $Λ$ -linear Ocneanu trace extends by scalar extension to ${tr}_{n}^{+} : H_{D} (n) \to D$ ; for $x \in H_{D} (n)$ it satisfies ${tr}_{n + 1}^{+} (x T_{n}) = z {tr}_{n}^{+} (x)$ and ${tr}_{n + 1}^{+} (x) = {tr}_{n}^{+} (x)$ (The Ocneanu Markov trace exists and is unique). Hence ${tr}_{n + 1}^{+} (x e_{n}) = \frac{z + 1}{v + 1} {tr}_{n}^{+} (x)$ ; at $z_{0} = - 1 / (v + 1)$ the factor is $λ = v / (v + 1)^{2}$ .

[F4]

$R$ has the universal property that unital ring homomorphisms $R \to T$ correspond to units $v_{0}, z_{0}, u_{0}, s_{0} \in T^{\times}$ with $s_{0}^{2} = v_{0}$ and $v_{0} z_{0} u_{0}^{2} = z_{0} + 1 - v_{0}$ , and $α = (u z)^{- 1}$ in $R$ (The HOMFLYPT coefficient ring).

[F5]

$P$ is the well-defined link invariant of The Hecke trace construction is an oriented link invariant, with $P (unknot) = 1$ , and it satisfies $l^{- 1} P_{+} - l P_{-} = m P_{0}$ for $l = u s$ , $m = s - s^{- 1}$ (The HOMFLYPT skein relation, The HOMFLYPT polynomial from the Hecke Markov trace).

[F6]

Literature input (quoted). The algebra $T L (n, τ)$ with generators $e_{1}, \dots, e_{n - 1}$ and relations $e_{i}^{2} = e_{i}$ , $e_{i} e_{i \pm 1} e_{i} = τ e_{i}$ , $e_{i} e_{j} = e_{j} e_{i}$ for $∣ i - j ∣ \geq 2$ carries a Markov trace normalized by $tr (1) = 1$ and $tr (x e_{n + 1}) = τ tr (x)$ for $x \in T L (n + 1, τ)$ (Jones, The Jones Polynomial, printed pp. 7-9). Birman--Brendle §4.3 defines the normalized Hecke-trace invariant $P (l, m)$ (their equations preceding (18)) and asserts in property 6 that $P (t, t^{1 / 2} - t^{- 1 / 2})$ is the Jones polynomial. Their equation (18) holds for arbitrary oriented skein triples. These are quoted source results for that specific trace construction, rather than an extension of the local braided-triple theorem or reliance on the inconsistent skein formula printed in Jones’ survey, p. 2.

Proof

1.1F1algebra

Idempotents. In $H_{D} (n)$ , $v + 1$ is a unit. From $T_{i}^{2} = (v - 1) T_{i} + v$ of [F1], $(T_{i} + 1)^{2} = (v + 1) (T_{i} + 1)$ ; dividing by $(v + 1)^{2}$ gives $e_{i}^{2} = e_{i}$ . The far-commutation $e_{i} e_{j} = e_{j} e_{i}$ for $∣ i - j ∣ > 1$ is inherited from $T_{i} T_{j} = T_{j} T_{i}$ of [F1], since $e_{i}$ is a polynomial in $T_{i}$ with coefficients in $D$ .

1.2F1F2algebra

The three-strand relation. In $H_{D} (n)$ , expanding $(T_{i} + 1) (T_{i + 1} + 1) (T_{i} + 1)$ with [F1] gives $E_{n}^{(i)} + v (T_{i} + 1) = E_{n}^{(i)} + v (v + 1) e_{i}$ : the six standard-basis elements of the parabolic subgroup are $1, T_{i}, T_{i + 1}, T_{i} T_{i + 1}, T_{i + 1} T_{i}, T_{i} T_{i + 1} T_{i}$ , and substituting $T_{i}^{2} = (v - 1) T_{i} + v$ leaves exactly $v (T_{i} + 1)$ in addition. Since $(T_{i} + 1) (T_{i + 1} + 1) (T_{i} + 1) = (v + 1)^{3} e_{i} e_{i + 1} e_{i}$ , this reads $(v + 1)^{3} e_{i} e_{i + 1} e_{i} = E_{n}^{(i)} + v (v + 1) e_{i}$ ; in ${T L}_{D} (n)$ , where the class of $E_{n}^{(i)}$ is zero by [F2], it becomes $e_{i} e_{i + 1} e_{i} = \frac{v}{(v + 1)^{2}} e_{i} = λ e_{i}$ . The mirrored computation gives $e_{i + 1} e_{i} e_{i + 1} = λ e_{i + 1}$ .

1.3F4algebra

The specialization homomorphism. Take $T$ as in the definition and put $v_{0} : = v$ , $z_{0} : = - 1 / (v + 1)$ , $u_{0} : = s$ , $s_{0} : = s$ in $T$ . These are units by construction, $s_{0}^{2} = v = s^{2} = v_{0}$ , and $v_{0} z_{0} u_{0}^{2} = v \cdot (- 1 / (v + 1)) \cdot v = - v^{2} / (v + 1)$ while $z_{0} + 1 - v_{0} = (- 1 + (v + 1) (1 - v)) / (v + 1) = - v^{2} / (v + 1)$ ; by the universal property of [F4] there is a unique unital ring homomorphism $φ : R \to T$ with these values, and it sends $α = (u z)^{- 1}$ to $(s \cdot (- 1 / (v + 1)))^{- 1} = - (v + 1) / s = - (s^{2} + 1) / s$ .

2.1F1step 1.1step 1.2algebra

Normalized generators. In the scalar extension of ${T L}_{D} (n)$ adjoining $s$ with $s^{2} = v$ , put $f_{i} = (v + 1) s^{- 1} e_{i} = λ^{- 1 / 2} e_{i}$ . Then $f_{i}^{2} = λ^{- 1} e_{i} = λ^{- 1 / 2} f_{i} = δ f_{i}$ with $δ = λ^{- 1 / 2} = (v + 1) s^{- 1} = s + s^{- 1}$ , and $f_{i} f_{i + 1} f_{i} = λ^{- 3 / 2} e_{i} e_{i + 1} e_{i} = λ^{- 3 / 2} λ e_{i} = λ^{- 1 / 2} e_{i} = f_{i}$ , with the symmetric relation for $f_{i + 1} f_{i} f_{i + 1}$ ; far-commutation is inherited from step 1.1.

2.2F4F5step 1.3algebra

The Jones invariant. For $β \in B_{n}$ with $n \geq 1$ put $V (\hat{β}) : = φ (P (\hat{β}))$ ; this is exactly the displayed specialization because $φ (u) = s$ and $φ (v) = v = s^{2}$ , and $α$ to $- (s^{2} + 1) / s$ . Since $φ$ is a ring homomorphism and $P$ is an invariant of oriented links by [F5], $V$ is an invariant of oriented links with $V (unknot) = φ (1) = 1$ . Applying $φ$ to the skein relation of [F5] and using $φ (l) = s^{2}$ , $φ (m) = s - s^{- 1}$ gives $s^{- 2} V_{+} - s^{2} V_{-} = (s - s^{- 1}) V_{0}$ on the braided triples supplied there, in the variable $t = s^{2}$ .

2.3F3F6step 1.2algebra

The Markov normalization on the tower. Extend the trace by scalar extension as in [F3]. For $x \in H_{D} (n)$ , $e_{n} = (T_{n} + 1) / (v + 1)$ , so ${tr}_{n + 1}^{+} (x e_{n}) = \frac{1}{v + 1} ({tr}_{n + 1}^{+} (x T_{n}) + {tr}_{n + 1}^{+} (x)) = \frac{z + 1}{v + 1} {tr}_{n}^{+} (x)$ , and at $z = z_{0}$ the factor is $\frac{v}{(v + 1)^{2}} = λ$ . This matches the Markov normalization with parameter $τ = λ$ in [F6]; it does not assert descent of the Ocneanu trace to ${T L}_{D} (n)$ .

3.1F4F5F6step 1.3step 2.2algebra∎

Identification and arbitrary skein triples. The ring $T$ is $Z [s^{\pm 1}, (s^{2} + 1)^{- 1}]$ by eliminating $v = s^{2}$ , hence injects into $Q (s)$ . In the source normalization of [F6], set its Hecke parameter to $v = s^{2}$ and its rescaling parameter to $κ = s^{2}$ ; then its generator rescaling is $\sqrt{κ} = s$ and its trace parameter is $- (1 - v) / (1 - κ v) = - 1 / (v + 1) = z_{0}$ . Its strand factor is $- κ^{- 1 / 2} (1 - κ v) / (1 - v) = - (v + 1) / s$ , exactly step 1.3. Thus our $V$ is the image of the same source trace construction with $l = \sqrt{κ v} = s^{2}$ and $m = s - s^{- 1}$ . Property 6 identifies this with the Jones polynomial, and source equation (18) gives the stated relation for every oriented skein triple. The identities hold in $T$ since both sides lie in $T$ and its embedding into $Q (s)$ is injective. The mirror substitution is the source chirality rule $P (l^{- 1}, - m)$ , which here is $s \mapsto s^{- 1}$ .

Remarks

The rescaling in the definition is the correct one: with $f_{i} = λ^{- 1} e_{i}$ the three-strand relation would give $f_{i} f_{i + 1} f_{i} = λ^{- 1} f_{i}$ , not $f_{i}$ ; the square root $λ^{- 1 / 2}$ (equivalently $v^{1 / 2}$ ) is necessary, and it exists in the Jones specialization ring $T$ .
The classical normalization of the Temperley--Lieb loop value is $δ = s + s^{- 1}$ with $s^{2} = v$ , in agreement with $λ^{- 1 / 2} = (v + 1) / v^{1 / 2}$ .
The two-strand example The Jones specialization of a two-strand closure ↗ and the three-crossing skein example The Hecke trace skein calculation for a three-crossing braid ↗ compute $V$ explicitly from this definition.

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Sources

Vaughan F. R. Jones, The Jones Polynomial, section 3 (the Temperley-Lieb algebra, relations and Markov trace, printed pp. 7-9) (standard reference, not scraped)
Joan S. Birman and Tara E. Brendle, Braids: A Survey, section 1.4.2 (the Jones algebra as a Hecke quotient, printed pp. 8-9); section 4.3 equation (18) and property 6 (skein relation and Jones specialization, printed pp. 50-51 in the downloaded PDF) (standard reference, not scraped)