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 By circle we mean an and imbedded is in the sense of differential geometry. A link is a disjoint union of imbedded circles in A link is oriented if each connected component is oriented. We shall identify a link with its "picture in the plane".
The conceptual ideal of when two links are the same is called ambient isotopy. More precisely, two oriented links and are equivalent under ambient isotopy if there is an orientation preserving diffeomorphism of which takes to In terms of pictures in the plane and are equivalent under ambient isotopy if the picture for can be transformed into the picture for by a sequence of Reidemeister moves:
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 be a set. An oriented link invariant with values in is a map
from the set of equivalence classes of oriented links under ambient isotopy to
There exists a unique oriented link invariant such that
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 and as follows:
A braid on consists of two rows of dots each, one above the other, and strands in such that
Composition of two braids on is given by identifying the bottom points of with the top points of The following are braids on 6 strands,
and the product is the braid
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 is the group of braids on strands and it is a famous theorem of E. Artin that has a presentation by generators
for and relations
It will be convenient to "orient" the strands of a braid so that they "travel" from top to bottom.
The closure of a braid on is the oriented link obtained by joining together (identifying) each dot in the top row to the corresponding dot in the bottom row. If
(Alexander) Every oriented link is the closure of a braid for some
The braid group can be embedded into the braid group by adding a strand.
Two braids and are Markov equivalent if they are equivalent under the equivalence relation on (disjoint union of which is defined by the relations
where in the relation (M2) the products and are obtained by viewing as an element of under the imbedding
(Markov) Two braids and have equivalent closures and (under ambient isotopy) if and only if and are Markov equivalent.
Let be a finite dimensional complex simple Lie algebra and let be the corresponding Drinfel'd-Jimbo quantum group. Let be the element of such that for all simple roots see II (2.6).
Let be a finite dimensional module. The quantum dimension of is
If then the quantum trace of is
Let be the irreducible of highest weight as given in VI (1.3) and VI (2.3). Then
is the half sum of the positive roots, and the inner product on is as given in II (2.7).
Recall that is a quasitriangular Hopf algebra and that therefore the category of finite dimensional is a braided SRMCwMFF. Let
be the braiding isomorphism from to It follows from the identity I (3.5) that the map
is well defined and that for all braids
Let be a finite dimensional complex simple Lie algebra and let be the corresponding Drinfel'd-Jimbo quantum group. Let be an irreducible of highest weight (see VI (1.3) and VI (2.3)). Let be the half sum of the positive roots and let be the inner product on as given in II (2.7). For each braid on define
where Then is a well defined link invariant.
The above theorem gives the Jones polynomial when the simple Lie algebra corresponding to the Dynkin diagram and is chosen to be the irreducible representation of with highest weight
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.