XI. Link invariants from quantum groups

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

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).

Knotes, links and isotopy

A knot is an embedded circle in 3. By circle we mean an S1 and imbedded is in the sense of differential geometry. A link is a disjoint union of imbedded circles in 3. 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 L1 and L2 are equivalent under ambient isotopy if there is an orientation preserving diffeomorphism of 3 which takes L1 to L2. In terms of pictures in the plane L1 and L2 are equivalent under ambient isotopy if the picture for L1 can be transformed into the picture for L2 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.

Link invariants

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


from the set of equivalence classes of oriented links under ambient isotopy to S.

There exists a unique oriented link invariant P: [ x,x-1, y,y-1 ] 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.


A braid on m-strands consists of two rows of m dots each, one above the other, and m strands in 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 b1, b2 on m-strands is given by identifying the bottom points of b1 with the top points of b2. The following are braids on 6 strands,

b1= ,b2= ,

and the product b1b2 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 m is the group of braids on m strands and it is a famous theorem of E. Artin that m has a presentation by generators

gi= 1 2 i-1 i i+1 i+2 m-1 m ,

for 1im-1, and relations

gigj= gjgi, if i-j>1, gigi+1gi= gi+1gi gi+1, for1im-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 (β^,m) of a braid βm on m-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 (β^,m) of a braid βm for some m.

Markov equivalence

The braid group m can be embedded into the braid group m+1 by adding a strand.

6 7

Two braids β1m and β2n are Markov equivalent if they are equivalent under the equivalence relation on mm (disjoint union of m) which is defined by the relations

  1. (M1) β~ββ β-1,for all β,βk, and
  2. (M2)β~β gk~βgk-1, ifβk;

where in the relation (M2) the products βgk and βgk-1 are obtained by viewing β as an element of k+1 under the imbedding kk+1.

(Markov) Two braids β1m and β2n have equivalent closures (β^1,m) and (β^2,n) (under ambient isotopy) if and only if β1 and β2 are Markov equivalent.

Quantum dimensions and quantum traces

Let 𝔤 be a finite dimensional complex simple Lie algebra and let 𝔘h𝔤 be the corresponding Drinfel'd-Jimbo quantum group. Let ρ~ be the element of 𝔥 such that αi(ρ~) =1 for all simple roots αi, see II (2.6).

Let V be a finite dimensional 𝔘h𝔤 module. The quantum dimension of V is

dimq(V)= TrV(ehρ~) .

If zEnd𝔘h𝔤(V) then the quantum trace of z is

trq(z)= TrV(ehρ~z) .

Let L(λ) be the irreducible 𝔘h𝔤-module of highest weight λ 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,

ρ=12α>0α is the half sum of the positive roots, and the inner product (,) on 𝔥* is as given in II (2.7).

Quantum traces give us link invariants!

Recall that 𝔘h𝔤 is a quasitriangular Hopf algebra and that therefore the category of finite dimensional 𝔘h𝔤-modules is a braided SRMCwMFF. Let


be the braiding isomorphism from VV to VV. It follows from the identity I (3.5) that the map

Φ: m End𝔘h𝔤 (Vm) gi Ri= id(i-1) RVV idm-(i+1)

is well defined and that Φ(β1β2) =Φ(β1) Φ(β2) for all braids β1,β2m.

Let 𝔤 be a finite dimensional complex simple Lie algebra and let 𝔘h𝔤 be the corresponding Drinfel'd-Jimbo quantum group. Let L(λ) be an irreducible 𝔘h𝔤-module of highest weight λ (see VI (1.3) and VI (2.3)). Let ρ be the half sum of the positive roots and let (,) be the inner product on 𝔥 as given in II (2.7). For each braid β on m-strands define

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

where q=eh. Then P is a well defined link invariant.

The above theorem gives the Jones polynomial when 𝔤=𝔰𝔩2, the simple Lie algebra corresponding to the Dynkin diagram A1, and L(λ) is chosen to be the irreducible representation of 𝔘h𝔤 with highest weight λ=ω1.

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.

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