XI. Link invariants from quantum groups

Last update: 14 October 2012

The theorems of Alexander and Markov given in (1.4) and (1.5) are considered classifical, they can be found in [Bir1976] Theorem 2.1 and Theorem 2.3, respectively. A sketch, with further references, of the proof of Theorem (1.7) can be found in [Cpr1994] 15.2. See [Jon1987] Prop. 6.2 for the proof of Theorem (1.2) and [Ste1994] Lemma 2.5 for the proof of Proposition (1.6).

A knot is an embedded circle in ${ℝ}^{3}\text{.}$ By circle we mean an ${S}^{1}$ and imbedded is in the sense of differential geometry. A link is a disjoint union of imbedded circles in ${ℝ}^{3}\text{.}$ A link is oriented if each connected component is oriented. We shall identify a link with its "picture in the plane".

$knot (unknot) knot (trefoil) knot (Borromean rings)$

The conceptual ideal of when two links are the same is called ambient isotopy. More precisely, two oriented links ${L}_{1}$ and ${L}_{2}$ are equivalent under ambient isotopy if there is an orientation preserving diffeomorphism of ${ℝ}^{3}$ which takes ${L}_{1}$ to ${L}_{2}\text{.}$ In terms of pictures in the plane ${L}_{1}$ and ${L}_{2}$ are equivalent under ambient isotopy if the picture for ${L}_{1}$ can be transformed into the picture for ${L}_{2}$ by a sequence of Reidemeister moves:

$(R1) ⟷ ⟷ (R2) ⟷ ⟷ (R3) ⟷$

These moves are applied locally to a region in the picture and all possible orientations of the strings are allowed. The equivalence relation on pictures in the plane gotten by only allowing moves (R2) and (R3) is called regular isotopy.

Let $S$ be a set. An oriented link invariant with values in $S$ is a map

$P:ℒ⟶S$

from the set $ℒ$ of equivalence classes of oriented links under ambient isotopy to $S\text{.}$

There exists a unique oriented link invariant $P:\phantom{\rule{0.2em}{0ex}}ℒ⟶ℤ\left[x,{x}^{-1},y,{y}^{-1}\right]$ such that

$P ( ) =1,andxP ( ) -x-1P ( ) =yP ( ) .$

The unusual notation in the second relation indicates changes to the link in a local region.

The link invariant defined in the above Theorem is the HOMFLY polynomial. Other famous link invariants can be obtained in a similar fashion by specializing $x$ and $y,$ as follows:

$Jones polynomial x=t-1 and y=t1/2- t-1/2, Conway polynomial x=1 and y=y, Alexander polynomial x=1 and y=t1/2- t-1/2.$

Braids

A braid on $m\text{-strands}$ consists of two rows of $m$ dots each, one above the other, and $m$ strands in ${ℝ}^{3}$ such that

1. each strand connects a dot in the top row to a dot in the bottom row,
2. the strands do no intersect,
3. every dot is incident to exactly one strand.

Composition of two braids ${b}_{1},$ ${b}_{2}$ on $m\text{-strands}$ is given by identifying the bottom points of ${b}_{1}$ with the top points of ${b}_{2}\text{.}$ The following are braids on 6 strands,

$b1= ,b2= ,$

and the product ${b}_{1}{b}_{2}$ is the braid

$b1b2= ,$

One should note that it is important to be careful in defining the word "strand" since the diagram



is not a legal braid.

The braid group ${ℬ}_{m}$ is the group of braids on $m$ strands and it is a famous theorem of E. Artin that ${ℬ}_{m}$ has a presentation by generators

$gi= … … 1 2 i-1 i i+1 i+2 m-1 m ,$

for $1\le i\le m-1,$ and relations

$gigj= gjgi, if ∣i-j∣>1, gigi+1gi= gi+1gi gi+1, for1≤i≤m-2.$

Every link is the closure of a braid

It will be convenient to "orient" the strands of a braid so that they "travel" from top to bottom.



The closure $\left(\stackrel{^}{\beta },m\right)$ of a braid $\beta \in {ℬ}_{m}$ on $m\text{-strands}$ is the oriented link obtained by joining together (identifying) each dot in the top row to the corresponding dot in the bottom row. If

$β= ,then (β^,3)= ,$

and if

$β= ,then (β^,3)= .$

(Alexander) Every oriented link is the closure $\left(\stackrel{^}{\beta },m\right)$ of a braid $\beta \in {ℬ}_{m}$ for some $m\text{.}$

Markov equivalence

The braid group ${ℬ}_{m}$ can be embedded into the braid group ${ℬ}_{m+1}$ by adding a strand.

$ℬ6 ⟶ ℬ7 ⟼$

Two braids ${\beta }_{1}\in {ℬ}_{m}$ and ${\beta }_{2}\in {ℬ}_{n}$ are Markov equivalent if they are equivalent under the equivalence relation on ${\bigsqcup }_{m}{ℬ}_{m}$ (disjoint union of ${ℬ}_{m}\text{)}$ which is defined by the relations

1. $\text{(M1)}\phantom{\rule{1em}{0ex}}{\beta }^{\prime }~\beta {\beta }^{\prime }{\beta }^{-1},\phantom{\rule{1em}{0ex}}\text{for all}\phantom{\rule{0.2em}{0ex}}\beta ,{\beta }^{\prime }\in {ℬ}_{k},\phantom{\rule{0.2em}{0ex}}\text{and}$
2. $\text{(M2)}\phantom{\rule{1em}{0ex}}\beta ~\beta {g}_{k}~\beta {g}_{k}^{-1},\phantom{\rule{2em}{0ex}}\text{if}\phantom{\rule{0.2em}{0ex}}\beta \in {ℬ}_{k};$

where in the relation (M2) the products $\beta {g}_{k}$ and $\beta {g}_{k}^{-1}$ are obtained by viewing $\beta$ as an element of ${ℬ}_{k+1}$ under the imbedding ${ℬ}_{k}\subseteq {ℬ}_{k+1}\text{.}$

(Markov) Two braids ${\beta }_{1}\in {ℬ}_{m}$ and ${\beta }_{2}\in {ℬ}_{n}$ have equivalent closures $\left({\stackrel{^}{\beta }}_{1},m\right)$ and $\left({\stackrel{^}{\beta }}_{2},n\right)$ (under ambient isotopy) if and only if ${\beta }_{1}$ and ${\beta }_{2}$ are Markov equivalent.

Quantum dimensions and quantum traces

Let $𝔤$ be a finite dimensional complex simple Lie algebra and let ${𝔘}_{h}𝔤$ be the corresponding Drinfel'd-Jimbo quantum group. Let $\stackrel{~}{\rho }$ be the element of $𝔥$ such that ${\alpha }_{i}\left(\stackrel{~}{\rho }\right)=1$ for all simple roots ${\alpha }_{i},$ see II (2.6).

Let $V$ be a finite dimensional ${𝔘}_{h}𝔤$ module. The quantum dimension of $V$ is

$dimq(V)= TrV(ehρ~) .$

If $z\in {\text{End}}_{{𝔘}_{h}𝔤}\left(V\right)$ then the quantum trace of $z$ is

$trq(z)= TrV(ehρ~z) .$

Let $L\left(\lambda \right)$ be the irreducible ${𝔘}_{h}𝔤\text{-module}$ of highest weight $\lambda$ as given in VI (1.3) and VI (2.3). Then

$dimq(L(λ)) =∏α>0 1-q(λ+ρ,α) 1-q(ρ,α) ,whereq=eh,$

$\rho =\frac{1}{2}\sum _{\alpha >0}\alpha$ is the half sum of the positive roots, and the inner product $\left(,\right)$ on ${𝔥}_{ℝ}^{*}$ is as given in II (2.7).

Quantum traces give us link invariants!

Recall that ${𝔘}_{h}𝔤$ is a quasitriangular Hopf algebra and that therefore the category of finite dimensional ${𝔘}_{h}𝔤\text{-modules}$ is a braided SRMCwMFF. Let

$R∨VV: V⊗V⟶V⊗V$

be the braiding isomorphism from $V\otimes V$ to $V\otimes V\text{.}$ It follows from the identity I (3.5) that the map

$Φ: ℬm ⟶ End𝔘h𝔤 (V⊗m) gi ⟼ R∨i= id⊗(i-1)⊗ R∨VV⊗ id⊗m-(i+1)$

is well defined and that $\Phi \left({\beta }_{1}{\beta }_{2}\right)=\Phi \left({\beta }_{1}\right)\Phi \left({\beta }_{2}\right)$ for all braids ${\beta }_{1},{\beta }_{2}\in {ℬ}_{m}\text{.}$

Let $𝔤$ be a finite dimensional complex simple Lie algebra and let ${𝔘}_{h}𝔤$ be the corresponding Drinfel'd-Jimbo quantum group. Let $L\left(\lambda \right)$ be an irreducible ${𝔘}_{h}𝔤\text{-module}$ of highest weight $\lambda$ (see VI (1.3) and VI (2.3)). Let $\rho$ be the half sum of the positive roots and let $\left(,\right)$ be the inner product on ${𝔥}_{ℝ}$ as given in II (2.7). For each braid $\beta$ on $m\text{-strands}$ define

$P(β^,m)= ( 1 q ⟨ λ,λ+2ρ ⟩ dimq(V) ) m trq (Φ(β)),$

where $q={e}^{h}\text{.}$ Then $P$ is a well defined link invariant.

The above theorem gives the Jones polynomial when $𝔤={𝔰𝔩}_{2},$ the simple Lie algebra corresponding to the Dynkin diagram ${A}_{1},$ and $L\left(\lambda \right)$ is chosen to be the irreducible representation of ${𝔘}_{h}𝔤$ with highest weight $\lambda ={\omega }_{1}\text{.}$

Notes and References

This is an excerpt from a paper entitled Quantum groups: A survey of definitions, motivations and results by Arun Ram. Research and writing supported in part by an Australian Research Council fellowship and a National Science Foundation grant DMS-9622985.