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The Jones specialization of a two-strand closure

Example

Assume AC for the link-invariance and closure-isotopy assertions below. For the two-strand braid $β = σ_{1}^{3} \in B_{2}$ the Ocneanu trace gives ${tr}_{2} (π (β)) = (v^{2} - v + 1) z + v (v - 1)$ (Birman--Brendle, Example 4.1 in their variables), and at the Jones specialization $z_{0} = - 1 / (v + 1)$ , $u = s$ , $t = s^{2}$ of The Temperley-Lieb quotient and the Jones specialization the value is $V (\hat{σ_{1}^{3}}) = - s^{8} + s^{6} + s^{2} = - t^{4} + t^{3} + t .$ This is the value of the invariant $V$ on the closure of $σ_{1}^{3}$ , which is the right-handed trefoil; it agrees with the trefoil value displayed in Jones’ survey, printed p. 1. Replacing $t$ by $t^{- 1}$ gives the value of the mirror trefoil in this same convention. The value satisfies $V (unknot) = 1$ and, on every skein triple, the Jones skein relation $s^{- 2} V_{+} - s^{2} V_{-} = (s - s^{- 1}) V_{0}$ of The Temperley-Lieb quotient and the Jones specialization. The three-crossing braid $σ_{1} σ_{2} σ_{1}$ is not a second representative of this knot: its closure is the Hopf link with value $- s^{5} - s$ , as computed in The Hecke trace skein calculation for a three-crossing braid.

Verification

Given: AC (The Axiom of Choice), the braid $σ_{1}^{3} \in B_{2}$ , the generator $T_{1} \in H (2)$ , the Ocneanu trace and the Jones specialization $V$ . AC is used through the cited link-invariance and closure-isotopy results; the trace computations are algebraic.

[A1] $T_{1}^{2} = (v - 1) T_{1} + v$ in $H (2)$ , and $π_{2} (σ_{1}^{3}) = T_{1}^{3}$ , $e (σ_{1}^{3}) = 3$ (The generic type-A Hecke algebra, The Hecke generators satisfy the Artin relations and are units, The exponent sum of a braid).

[A2] ${tr}_{2} (T_{1}) = z$ , ${tr}_{2} (1) = 1$ , and ${tr}_{2}$ is $Λ$ -linear (The Ocneanu Markov trace exists and is unique).

[A3] $V (\hat{β}) = s^{e (β)} (- \frac{s^{2} + 1}{s})^{n - 1} {tr}_{n} (π_{n} (β)) ∣_{z = z_{0}}$ for $β \in B_{n}$ , with $z_{0} = - 1 / (v + 1)$ and $s^{2} = v$ ; $V$ is an oriented link invariant satisfying the Jones skein relation and $V (unknot) = 1$ (The Temperley-Lieb quotient and the Jones specialization, The HOMFLYPT skein relation, The HOMFLYPT polynomial from the Hecke Markov trace).

[A4] The closure of $σ_{1}^{3}$ is the right-handed trefoil; Jones’ survey, printed p. 1, displays the trefoil value $t + t^{3} - t^{4}$ . Its mirror rule replaces $t$ by $t^{- 1}$ , so the mirror $\hat{σ_{1}^{- 3}}$ has value $t^{- 1} + t^{- 3} - t^{- 4}$ in this convention (Jones, printed pp. 1 and 4).

[A5] Conjugation and positive stabilization preserve the isotopy class of the closure, the closure of $σ_{1}^{2} \in B_{2}$ has two components, and $σ_{1}^{- 1} (σ_{1}^{2} σ_{2}) σ_{1} = σ_{1} σ_{2} σ_{1}$ (Markov conjugation and stabilization moves, Markov moves preserve the oriented closure up to isotopy, The closure of a geometric braid).

Proof technique: direct computation from the quadratic relation.

1.1A1A2algebra

The trace of $σ_{1}^{3}$ . By [A1] and $T_{1}^{2} = (v - 1) T_{1} + v$ , $T_{1}^{3} = T_{1} T_{1}^{2} = (v - 1) T_{1}^{2} + v T_{1} = (v - 1) ((v - 1) T_{1} + v) + v T_{1} = ((v - 1)^{2} + v) T_{1} + v (v - 1) = (v^{2} - v + 1) T_{1} + v (v - 1)$ . By linearity and [A2], ${tr}_{2} (T_{1}^{3}) = (v^{2} - v + 1) z + v (v - 1)$ , the displayed trace value.

2.1A1A2A3step 1.1algebra

The Jones value. At $z_{0} = - 1 / (v + 1)$ , ${tr}_{2} (T_{1}^{3}) ∣_{z_{0}} = \frac{- (v^{2} - v + 1) + v (v - 1) (v + 1)}{v + 1} = \frac{v^{3} - v^{2} - 1}{v + 1}$ . By [A3] with $n = 2$ and $e (σ_{1}^{3}) = 3$ , $V (\hat{σ_{1}^{3}}) = s^{3} (- \frac{s^{2} + 1}{s}) \frac{v^{3} - v^{2} - 1}{v + 1} = - s^{2} (s^{2} + 1) \frac{s^{6} - s^{4} - 1}{s^{2} + 1} = - s^{2} (s^{6} - s^{4} - 1) = - s^{8} + s^{6} + s^{2}$ , i.e. $- t^{4} + t^{3} + t$ for $t = s^{2}$ .

3.1A3A4step 2.1

Normalization and skein. $V (unknot) = 1$ and the skein relation $s^{- 2} V_{+} - s^{2} V_{-} = (s - s^{- 1}) V_{0}$ hold by [A3]; in the variable $t = s^{2}$ this is $t^{- 1} V_{+} - t V_{-} = (t^{1 / 2} - t^{- 1 / 2}) V_{0}$ . The closure of $σ_{1}^{3}$ is the right-handed trefoil by [A4], and applying $t \mapsto t^{- 1}$ to the value $- t^{4} + t^{3} + t$ of step 2.1 gives $- t^{- 4} + t^{- 3} + t^{- 1}$ , the mirror polynomial from [A4]. The value of step 2.1 itself agrees with the trefoil table value in the cited survey; no inverse-variable table comparison is needed.

4.1A1A2A3A5step 2.1∎

The cross-check. For the $2$ -braid representative $σ_{1}^{2}$ one computes ${tr}_{2} (T_{1}^{2}) = (v - 1) z + v$ from [A1] and [A2], so at $z_{0}$ its value is $(v^{2} + 1) / (v + 1)$ and by [A3] $V (\hat{σ_{1}^{2}}) = s^{2} (- \frac{s^{2} + 1}{s}) \frac{v^{2} + 1}{v + 1} = - s (s^{4} + 1) = - s^{5} - s$ , which is different from $- s^{8} + s^{6} + s^{2}$ ; by [A5] the closure of $σ_{1} σ_{2} σ_{1}$ is the closure of that positive stabilization of $σ_{1}^{2}$ , so $σ_{1} σ_{2} σ_{1}$ is not a second braid representative of the trefoil and the two computations are consistent with the invariance of [A3].

Remarks

The trace relation $(T_{1} - q) (T_{1} + 1) = 0$ of the finite Hecke algebra gives $tr (σ_{1}^{3}) = (q^{2} - q + 1) z + q (q - 1)$ in the variables of Birman--Brendle Example 4.1, which is the displayed value with $q = v$ .
The half-integer powers appearing in the Hopf-link value of the cross-check reflect the two components of that link; the trefoil is a knot, so its specialization lies in $s^{2} Z [s^{\pm 2}]$ , as the value $- s^{8} + s^{6} + s^{2}$ shows.

Depends on

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Sources

Joan S. Birman and Tara E. Brendle, Braids: A Survey, Example 4.1 (printed p. 49) and section 4.3 property 6 (the Jones specialization, printed p. 51) (standard reference, not scraped)
Vaughan F. R. Jones, The Jones Polynomial, Introduction: trefoil example on printed p. 1 and mirror substitution on printed p. 4 (standard reference, not scraped)