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The HOMFLYPT skein relation

Statement

Assume the Axiom of Choice. Let $P$ be the oriented link invariant of The Hecke trace construction is an oriented link invariant, with coefficient ring $R$ and variables $l = u s$ , $m = s - s^{- 1}$ , $v = s^{2}$ as in The HOMFLYPT coefficient ring. Let $x, y$ be Artin words in the generators of $B_{n}$ and $1 \leq i \leq n - 1$ , and let $L_{+}, L_{-}, L_{0}$ be the oriented links represented by the closures of $x σ_{i} y$ , $x σ_{i}^{- 1} y$ and $x y$ ; these three braid words differ only at one crossing between the strands $i, i + 1$ , so the three link diagrams form a skein triple. Then $l^{- 1} P (L_{+}) - l P (L_{-}) = m P (L_{0}),$ and $P (unknot) = 1$ . Equivalently, in the normalisation of the trace tower, $u^{- 1} P_{+} - v u P_{-} = (v - 1) P_{0} .$

Facts & Assumptions

Given: AC (The Axiom of Choice), the invariant $P$ of The Hecke trace construction is an oriented link invariant, a braid word $x σ_{i} y$ in $B_{n}$ and the corresponding skein triple $(L_{+}, L_{-}, L_{0})$ . The link-invariance assertion for $P$ uses AC as recorded in its supplier; the skein computation itself is algebraic.

[F1]

$P (L_{+}) = u^{e (x) + e (y) + 1} α^{n - 1} {tr}_{n} (π_{n} (x σ_{i} y))$ , $P (L_{-}) = u^{e (x) + e (y) - 1} α^{n - 1} {tr}_{n} (π_{n} (x σ_{i}^{- 1} y))$ and $P (L_{0}) = u^{e (x) + e (y)} α^{n - 1} {tr}_{n} (π_{n} (x y))$ , since all three words lie in $B_{n}$ and $α = (u z)^{- 1}$ (The HOMFLYPT polynomial from the Hecke Markov trace, The exponent sum of a braid).

[F2]

$T_{i} = v T_{i}^{- 1} + (v - 1)$ for every generator, and $π_{n}$ is multiplicative on words, so in $H (n)$ $π_{n} (x σ_{i} y) = v π_{n} (x σ_{i}^{- 1} y) + (v - 1) π_{n} (x y)$ (The Markov trace of an inverse Hecke generator, The Hecke generators satisfy the Artin relations and are units).

[F3]

${tr}_{n}$ is $Λ$ -linear, so applying it to the identity of [F2] gives the corresponding relation between the three trace values (The Ocneanu Markov trace exists and is unique).

[F4]

$l = u s$ , $m = s - s^{- 1}$ , $s^{2} = v$ , and $(v - 1) s^{- 1} = s - s^{- 1} = m$ in $R$ (The HOMFLYPT coefficient ring).

[F5]

$P (unknot) = 1$ (The Hecke trace construction is an oriented link invariant), and the closures of the three words represent the oriented links $L_{+}, L_{-}, L_{0}$ of the statement (The closure of a geometric braid).

Proof

1.1F1F2F3

The trace identity. By [F2] and the $Λ$ -linearity of the trace [F3], $A : = v B + (v - 1) C$ where $A = {tr}_{n} (π_{n} (x σ_{i} y))$ , $B = {tr}_{n} (π_{n} (x σ_{i}^{- 1} y))$ and $C = {tr}_{n} (π_{n} (x y))$ are the three trace values of [F1].

2.1F1step 1.1algebra

Normalisation. Multiply the identity of step 1.1 by $u^{e (x) + e (y)} α^{n - 1}$ and use [F1]: $u^{- 1} P (L_{+}) = v u P (L_{-}) + (v - 1) P (L_{0})$ , i.e. $u^{- 1} P_{+} - v u P_{-} = (v - 1) P_{0}$ , the second displayed relation.

3.1F4F5step 2.1algebra∎

The $(l, m)$ form. Divide the identity of step 2.1 by $s$ and use [F4]: $u^{- 1} s^{- 1} P_{+} - v u s^{- 1} P_{-} = (v - 1) s^{- 1} P_{0}$ ; here $u^{- 1} s^{- 1} = (u s)^{- 1} = l^{- 1}$ , $v u s^{- 1} = s^{2} u s^{- 1} = u s = l$ because $s^{2} s^{- 1} = s$ , and $(v - 1) s^{- 1} = s - s^{- 1} = m$ . Hence $l^{- 1} P_{+} - l P_{-} = m P_{0}$ , which is the first displayed relation; the normalization $P (unknot) = 1$ is [F5].

Remarks

The proof uses only the quadratic Hecke relation $T_{i} = v T_{i}^{- 1} + (v - 1)$ and the linearity of the trace; no reduced or unreduced Burau matrix enters the skein relation, which is why the invariant is defined for all braids.
The variable dictionary is $l = u s$ , $m = s - s^{- 1}$ , $u^{2} = z_{-} / z$ ; substituting $z = z_{0} = - 1 / (v + 1)$ , $u = s$ turns the relation into the Jones skein relation of The Temperley-Lieb quotient and the Jones specialization.

Depends on

Used by

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Sources

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