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The HOMFLYPT polynomial from the Hecke Markov trace

Definition

Assume AC for the arbitrary-braid closure convention. Let $R$ be the coefficient ring of The HOMFLYPT coefficient ring with its elements $v, z, s, u, l, m$ and the relations $s^{2} = v$ , $u^{2} = z_{-} / z$ , $l = u s$ , $m = s - s^{- 1}$ , and let $β \in B_{n}$ with closure $\hat{β}$ (The closure of a geometric braid). Write $e (β)$ for the exponent sum of The exponent sum of a braid, $π_{n} : B_{n} \to H (n)^{\times}$ for the homomorphism of The Hecke generators satisfy the Artin relations and are units, and ${tr}_{n}$ for the Ocneanu trace of The Ocneanu Markov trace exists and is unique. The HOMFLYPT polynomial from the Hecke Markov trace is $P (\hat{β}) : = u^{e (β)} α^{n - 1} {tr}_{n} (π_{n} (β)) \in R, α : = (u z)^{- 1},$ for $n \geq 1$ . For the empty link, the closure of the unique braid in $B_{0}$ , set $P (\emptyset) : = α^{- 1} = u z$ separately; no ${tr}_{0}$ is used. In the localization $R [m^{- 1}]$ , the identity $l^{- 1} - l = m α$ from The HOMFLYPT coefficient ring gives $α = \frac{l^{- 1} - l}{m}, u = l s^{- 1} .$ Thus the image of $P (\hat{β})$ in $R [m^{- 1}]$ is $u^{e (β)} (\frac{l^{- 1} - l}{m})^{n - 1} {tr}_{n} (π_{n} (β)) .$

Well-formedness. $e (β)$ and $π_{n} (β)$ depend only on the braid element, not on the chosen Artin word, and ${tr}_{n} (π_{n} (β))$ lies in $Λ$ and is mapped to $R$ by the coefficient-ring structure map; $u$ and $α$ are elements of $R$ , with $u$ a unit, so the displayed product is a well-defined element of $R$ .

Caveats. The construction as displayed is a function on braids on a fixed number of strands; it depends on the braid representative a priori, and the statement that it is independent of the representative of an oriented link is the content of The Hecke trace construction is an oriented link invariant ↗, not of this definition. The coefficient ring $R$ is the formal localised ring of The HOMFLYPT coefficient ring; the normalisation $α = (u z)^{- 1}$ is the one that makes the two Markov stabilisations scale by the same factor, as proved in clauses (2)-(3) of that invariance theorem.

Facts & Assumptions

Given: AC (The Axiom of Choice), the coefficient ring $R$ of The HOMFLYPT coefficient ring, an integer $n \geq 1$ , a braid $β \in B_{n}$ and its closure $\hat{β}$ . AC implies countable choice (AC implies DC implies countable choice), the hypothesis used by the arbitrary-braid closure convention in The closure of a geometric braid; the trace formula itself is algebraic.

[F1]

$R$ is a commutative $Λ = Z [v^{\pm 1}, z]$ -algebra; $v, z, u, s, l$ are units, $s^{2} = v$ , $u^{2} = z_{-} / z$ with $z_{-} = v^{- 1} (z + 1 - v)$ , $m = s - s^{- 1}$ , and $l^{- 1} - l = m (u z)^{- 1}$ (The HOMFLYPT coefficient ring). In $R [m^{- 1}]$ one may divide this identity by $m$ .

[F2]

The exponent sum $e : B_{n} \to Z$ is the unique homomorphism with $e (σ_{i}) = 1$ , and for every Artin word $β = σ_{i_{1}}^{ε_{1}} \dots σ_{i_{k}}^{ε_{k}}$ one has $e (β) = \sum_{r} ε_{r}$ independently of the word (The exponent sum of a braid).

[F3]

The assignment $σ_{i} \mapsto T_{i}$ induces a group homomorphism $π_{n} : B_{n} \to H (n)^{\times}$ with $π_{n} (σ_{i_{1}}^{ε_{1}} \dots σ_{i_{k}}^{ε_{k}}) = T_{i_{1}}^{ε_{1}} \dots T_{i_{k}}^{ε_{k}}$ for every Artin word (The Hecke generators satisfy the Artin relations and are units).

[F4]

The Ocneanu trace is a family of $Λ$ -linear maps ${tr}_{n} : H (n) \to Λ$ satisfying (M1)--(M4), and it satisfies ${tr}_{n + 1} (x T_{n} y) = z {tr}_{n} (x y)$ for all $x, y \in H (n)$ (The Ocneanu Markov trace exists and is unique); in particular ${tr}_{n} (π_{n} (β)) \in Λ$ , with its image used in $R$ .

[F5]

The closure $\hat{β}$ of a braid $β \in B_{n}$ is an oriented link in $S^{3}$ (The closure of a geometric braid).

Proof

1.1F1F2F3F4

Well-formedness of the factors. By [F2] the integer $e (β)$ depends only on the element $β$ ; by [F3] the element $π_{n} (β) \in H (n)^{\times}$ depends only on $β$ and not on the Artin word; by [F4] the trace of that element lies in $Λ$ and has a specified image in $R$ . The elements $u$ and $α = (u z)^{- 1}$ of $R$ exist because $u$ and $z$ are units of $R$ by [F1]. Hence the product $u^{e (β)} α^{n - 1} {tr}_{n} (π_{n} (β))$ is a well-defined element of $R$ , and it is computed from $β$ alone, not from a word.

1.2F1algebra

The identities in the $(l, m)$ variables. By [F1] one has $l^{- 1} - l = m (u z)^{- 1} = m α$ . In the localization $R [m^{- 1}]$ this gives $α = (l^{- 1} - l) / m$ . Also $l = u s$ and $s$ is a unit, so $u = l s^{- 1}$ . Substituting these into the definition gives the displayed formula for the image of $P$ in that localization.

2.1F1F2F3F5step 1.1∎

Dependence on the representative. The closure $\hat{β}$ is defined for every braid $β \in B_{n}$ by [F5]; the definition produces an element $P (\hat{β}) \in R$ for each braid, and no claim that two braids with isotopic closures give the same value is made here: that is exactly the statement proved in The Hecke trace construction is an oriented link invariant ↗. At $n = 0$ , the separate value $α^{- 1}$ exists because $u, z$ are units; no stabilization starts at $B_{0}$ .

Remarks

The coefficient form in $R [m^{- 1}]$ is used by The HOMFLYPT skein relation; the skein identity itself holds already in $R$ .
The normalisation $α = (u z)^{- 1}$ is forced by the two Markov moves: the positive stabilisation multiplies the trace by $z$ and the negative one by $z_{-}$ , and the two relations $u α z = 1$ and $u^{2} = z_{-} / z$ of The HOMFLYPT coefficient ring are precisely what make the two normalising factors $u α z$ and $u^{- 1} α z_{-}$ equal to $1$ ; see The Hecke trace construction is an oriented link invariant ↗.
The unknot is the closure of $1 \in B_{1}$ and has $P = 1$ because ${tr}_{1} (1) = 1$ by (M1); the empty product $n - 1 = 0$ contributes $α^{0} = 1$ .

Depends on

Used by

An unnormalized Hecke trace is not Markov invariant Counterexample
The Temperley-Lieb quotient and the Jones specialization Definition
The Hecke trace skein calculation for a three-crossing braid Example
The Jones specialization of a two-strand closure Example
Khovanov-Rozansky homology categorifies the HOMFLYPT polynomial Theorem
The Hecke trace construction is an oriented link invariant Theorem
The HOMFLYPT skein relation Theorem

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Sources

Theo Johnson-Freyd, MATH 448 Reshetikhin-Turaev invariants, lecture notes 15 January 2016, Sections 1-3 (the Ocneanu trace normalization and the HOMFLYPT construction) (standard reference, not scraped)
Joan S. Birman and Tara E. Brendle, Braids: A Survey, section 4.3 (printed pp. 47-49): the HOMFLYPT normalization from the Hecke trace (standard reference, not scraped)