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The Hecke trace skein calculation for a three-crossing braid

Example

Assume AC for the link-invariance and closure-isotopy assertions below. Work in the coefficient ring $R$ of The HOMFLYPT coefficient ring with the Ocneanu trace $tr$ of The Ocneanu Markov trace exists and is unique and the invariant $P$ of The HOMFLYPT polynomial from the Hecke Markov trace. For the three-crossing braid $β = σ_{1} σ_{2} σ_{1} \in B_{3}$ one has ${tr}_{3} (π_{3} (β)) = z (z (v - 1) + v) = z^{2} (v - 1) + z v, P (\hat{β}) = u^{3} α^{2} (z^{2} (v - 1) + z v) .$ With the trace values $A = {tr}_{3} (T_{1} T_{2} T_{1}), B = {tr}_{3} (T_{1} T_{2}^{- 1} T_{1}), C = {tr}_{3} (T_{1}^{2})$ one has $A = z^{2} (v - 1) + z v$ , $B = (1 - v^{- 1}) z^{2} + (3 - v - v^{- 1}) z + (1 - v)$ , $C = (v - 1) z + v$ , and the skein relation $l^{- 1} P (L_{+}) - l P (L_{-}) = m P (L_{0})$ of The HOMFLYPT skein relation holds on the triple $(L_{+}, L_{-}, L_{0}) = (\hat{σ_{1} σ_{2} σ_{1}}, \hat{σ_{1} σ_{2}^{- 1} σ_{1}}, \hat{σ_{1} σ_{1}})$ at the middle crossing ( $x = y = σ_{1}$ in $B_{3}$ ); the relation is equivalent to the polynomial identity $A - v B = (v - 1) C$ , both sides being $(v - 1)^{2} z + v (v - 1)$ .

The closure $\hat{β}$ is not a knot: the conjugation $σ_{1}^{- 1} (σ_{1}^{2} σ_{2}) σ_{1} = σ_{1} σ_{2} σ_{1}$ exhibits $σ_{1} σ_{2} σ_{1}$ as conjugate to $σ_{1}^{2} σ_{2}$ , the positive stabilization of $σ_{1}^{2} \in B_{2}$ , so $\hat{β}$ is isotopic to $\hat{σ_{1}^{2}}$ , the $(2, 2)$ -torus link, i.e. the Hopf link with two components (Markov conjugation and stabilization moves, Markov moves preserve the oriented closure up to isotopy). Consistently, at the Jones specialization $z_{0} = - 1 / (v + 1)$ , $u = s$ , $t = s^{2}$ of The Temperley-Lieb quotient and the Jones specialization the invariant takes the value $V (\hat{β}) = - s^{5} - s = - t^{1 / 2} (t^{2} + 1)$ on $\hat{β}$ , and the same value on the $2$ -braid representative $\hat{σ_{1}^{2}}$ ; this is the value of the Hopf link in the normalization of this page, with the convention fixed by The Temperley-Lieb quotient and the Jones specialization.

Verification

Given: AC (The Axiom of Choice), the braid $β = σ_{1} σ_{2} σ_{1} \in B_{3}$ , the generators $T_{1}, T_{2} \in H (3)$ , the Ocneanu trace and the invariant $P$ . AC is used only through the cited link-invariance and closure-isotopy results; the trace calculations are algebraic.

[A1] ${tr}_{n + 1} (x T_{n} y) = z {tr}_{n} (x y)$ and ${tr}_{n + 1} (x T_{n}^{- 1}) = z_{-} {tr}_{n} (x)$ for $x, y \in H (n)$ , ${tr}_{n} (1) = 1$ , ${tr}_{n} (T_{i}) = z$ and ${tr}_{n + 1} \circ ι_{n} = {tr}_{n}$ (The Ocneanu Markov trace exists and is unique, The Markov trace of an inverse Hecke generator).

[A2] $T_{i}^{2} = (v - 1) T_{i} + v$ and $T_{1} T_{2} T_{1} = T_{2} T_{1} T_{2}$ in $H (n)$ (The generic type-A Hecke algebra, The Markov trace on the type-A Hecke tower).

[A3] $P (\hat{β}) = u^{e (β)} α^{n - 1} {tr}_{n} (π_{n} (β))$ for $β \in B_{n}$ , the invariant is unchanged by Markov moves, $α = (u z)^{- 1}$ , $l = u s$ , $m = s - s^{- 1}$ in $R$ , and $l^{- 1} P_{+} - l P_{-} = m P_{0}$ (The HOMFLYPT polynomial from the Hecke Markov trace, The Hecke trace construction is an oriented link invariant, The HOMFLYPT skein relation).

[A4] $π_{n}$ is multiplicative with $π_{n} (σ_{i}) = T_{i}$ and $e (σ_{i_{1}}^{ε_{1}} \dots σ_{i_{k}}^{ε_{k}}) = \sum_{r} ε_{r}$ (The Hecke generators satisfy the Artin relations and are units, The exponent sum of a braid).

[A5] Conjugation of braids and positive stabilization preserve the isotopy class of the closure, and the closure of $σ_{1}^{2} \in B_{2}$ is the Hopf link (Markov conjugation and stabilization moves, Markov moves preserve the oriented closure up to isotopy, The closure of a geometric braid); the Jones specialization is $V = s^{e} (- \frac{s^{2} + 1}{s})^{n - 1} {tr}_{n} (π_{n} (β)) ∣_{z = z_{0}}$ with $z_{0} = - 1 / (v + 1)$ and $s^{2} = v$ (The Temperley-Lieb quotient and the Jones specialization).

Proof technique: direct computation from the trace recursion.

1.1A1A2A3A4algebra

The trace of the three-crossing braid. By [A4], $π_{3} (σ_{1} σ_{2} σ_{1}) = T_{1} T_{2} T_{1}$ ; by the recursion of [A1] with $n = 2$ and $x = y = T_{1}$ , ${tr}_{3} (T_{1} T_{2} T_{1}) = z {tr}_{2} (T_{1}^{2})$ . By [A2], $T_{1}^{2} = (v - 1) T_{1} + v$ , so by linearity and [A1], ${tr}_{2} (T_{1}^{2}) = (v - 1) z + v$ . Hence $A = {tr}_{3} (T_{1} T_{2} T_{1}) = z^{2} (v - 1) + z v$ , and $P (\hat{β}) = u^{e (β)} α^{2} A = u^{3} α^{2} (z^{2} (v - 1) + z v)$ since $e (β) = 3$ by [A4].

1.2A4A5givenalgebra

The closure is the Hopf link. The braid identity $σ_{1}^{- 1} (σ_{1}^{2} σ_{2}) σ_{1} = σ_{1} σ_{2} σ_{1}$ holds in $B_{3}$ by cancellation; since $σ_{1}^{2} \in B_{2}$ embeds as $σ_{1}^{2} \in B_{3}$ , the element $σ_{1}^{2} σ_{2}$ is a positive stabilization of $σ_{1}^{2}$ , and [A5] shows that the closures of $σ_{1} σ_{2} σ_{1}$ and $σ_{1}^{2}$ are ambient-isotopic. The endpoint permutation of $σ_{1}^{2}$ is $(1 2)^{2} = 1$ , so [A5] gives two components. Closing its two positive crossings yields the usual two-crossing positive Hopf diagram, namely the $(2, 2)$ -torus link. Hence $\hat{β}$ is that Hopf link. In particular $\hat{β}$ is not a knot, and the knot normalization is not used.

2.1A1A2step 1.1algebra

The other two trace values. The same recursion gives ${tr}_{3} (T_{1}^{2}) = {tr}_{2} (T_{1}^{2}) = C = (v - 1) z + v$ , and with $T_{2}^{- 1} = v^{- 1} T_{2} + (v^{- 1} - 1)$ from [A1], $B = {tr}_{3} (T_{1} T_{2}^{- 1} T_{1}) = v^{- 1} A + (v^{- 1} - 1) C = (1 - v^{- 1}) z^{2} + (3 - v - v^{- 1}) z + (1 - v)$ .

2.2A5step 1.1algebra

The Jones specialization. By [A5], with $z_{0} = - 1 / (v + 1)$ and $u = s$ , ${tr}_{3} (π_{3} (β)) ∣_{z_{0}} = - (v^{2} + 1) / (v + 1)^{2}$ and $V (\hat{β}) = s^{3} (- \frac{s^{2} + 1}{s})^{2} (- \frac{v^{2} + 1}{(v + 1)^{2}}) = - s (s^{4} + 1) = - s^{5} - s = - t^{1 / 2} (t^{2} + 1)$ with $t = s^{2}$ .

3.1A2A3A4step 1.1step 2.1algebra

The skein identity. Applying $P$ to the three words $x σ_{2} y = σ_{1} σ_{2} σ_{1}$ , $x σ_{2}^{- 1} y = σ_{1} σ_{2}^{- 1} σ_{1}$ and $x y = σ_{1}^{2}$ with $x = y = σ_{1}$ and using [A3] and [A4], $P (L_{+}) = u^{3} α^{2} A$ , $P (L_{-}) = u α^{2} B$ and $P (L_{0}) = u^{2} α^{2} C$ . The skein relation $l^{- 1} P_{+} - l P_{-} = m P_{0}$ of [A3] reduces, after cancelling the common factor $u^{2} α^{2}$ and multiplying by $s$ with $s^{2} = v$ , to $A - v B = (v - 1) C$ . Expanding, $A - v B = (v - 1) z^{2} + v z - v [(1 - v^{- 1}) z^{2} + (3 - v - v^{- 1}) z + (1 - v)] = (v - 1)^{2} z + v (v - 1) = (v - 1) C$ , so the relation holds identically in $Λ$ .

4.1A1A3A5step 1.2step 2.2algebra∎

Agreement with the two-strand representative. For $σ_{1}^{2} \in B_{2}$ one computes ${tr}_{2} (T_{1}^{2}) = C = (v - 1) z + v$ , so by [A5] $V (\hat{σ_{1}^{2}}) = s^{2} (- \frac{s^{2} + 1}{s}) \frac{v^{2} + 1}{v + 1} = - s (s^{4} + 1)$ , the same value as step 2.2, as the invariance of [A3] requires for two representatives of the same link. This completes the computation and the cross-check.

Remarks

The value $- t^{1 / 2} (t^{2} + 1)$ is the Jones value of the Hopf link in this page's convention; the half-integral power of $t$ reflects the two components of the link in the normalization used here.
The trace identity $A = v B + (v - 1) C$ of step 3.1 is the one used in The HOMFLYPT skein relation; the example exhibits it on a word in which the middle letter is isolated, so no other relation enters.

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Sources

Theo Johnson-Freyd, MATH 448 Reshetikhin-Turaev invariants, lecture notes 15 January 2016, Sections 1-3 (the trace recursion and the skein relation) (standard reference, not scraped)
Joan S. Birman and Tara E. Brendle, Braids: A Survey, section 4.3 (printed pp. 47-49): trace values and the Jones specialization of the HOMFLYPT polynomial (standard reference, not scraped)