Orthogonal Quantum Group Invariants of Links
Abstract.
We study the ChernSimons partition function of orthogonal quantum group invariants, and propose a new orthogonal LabastidaMariñoOoguriVafa conjecture as well as degree conjecture for free energy associated to the orthogonal ChernSimons partition function. We prove the degree conjecture and some interesting cases of orthogonal LMOV conjecture. In particular, We provide a formula of colored Kauffman polynomials for torus knots and links, and applied this formula to verify certain case of the conjecture at roots of unity except . We also derive formulas of LickorishMillett type for Kauffman polynomials and relate all these to the orthogonal LMOV conjecture.
1. Introduction
1.1. Overview
Jones’ seminal papers [18, 19] initiated a new era in knot theory. The HOMFLY polynomial [13] and Kauffman [23] polynomial for links were subsequently discovered. In the 1990’s, WittenReshetikhinTuraev constructed the colored version of these invariants, either by path integrals in physics [54], or by the representation theory of quantum groups [44, 45]. These works lead to a unified understanding of quantum group invariants of links.
The colored HOMFLY polynomials, which are associated to the special linear quantum groups, have been studied more carefully after physicists proposed a conjectural relationship between ChernSimons theory and GromovWitten invariants. The MariñoVafa formula and the topological vertex [1, 27, 30, 31] are examples illustrating this socalled string duality. The LabastidaMariñoOoguriVafa conjecture [25, 26, 36] gave highly nontrivial relations between colored HOMFLY polynomials. The first such relation is the classical LichorishMillett theorem [28]. The integers coefficients that appear in the LMOV conjecture are called the BPS numbers in string theory, and also related to the integrality in the GopakumarVafa conjecture [16] for GromovWitten invariants [39]. By using the cabling technique, XiaoSong Lin and Hao Zheng [29] obtained a formula for colored HOMFLY polynomials of torus links in terms of LittlewoodRichardson coefficients, and they were able to check certain cases of the LMOV conjecture for a few (small) torus knots and links. The LMOV conjecture was recently proved by Kefeng Liu and Pan Peng [32], based on the cabling technique and a careful degree analysis of the cutjoin equations.
Actually the LMOV conjecture is part of a bigger picture, the large duality, proposed by ’t Hooft [49] in the 1970’s. Large duality states that the duality between ChernSimons gauge theory of and topological string theory on the resolved conifold.
In mathematics, the LMOV conjecture predicts that the reformulated invariants (some combination) of colored HOMFLY/Kauffman polynomials are in the ring , where is the quantum deformation number. Through this way, these reformulated invariants has the similar expression as the original HOMFLY/Kauffman polynomials which has variables , and .
1.2. Orthogonal LabastidaMariñoOoguriVafa Conjecture
The study of colored Kauffman polynomials is more difficult. For instance, the definition of the ChernSimons partition function for the orthogonal quantum groups involves representations of the Brauer centralizer algebras, which admit more complicated orthogonal relations (see [40, 41, 42]). The orthogonal analog of cutjoin equation [30, 32] can be found in [10].
In this paper, we propose a new conjecture on the reformulated invariants, developed in collaboration with Nicolai Reshetikhin, which is the orthogonal quantum group analog of the original LMOV conjecture. Let be a link with components and let be the orthogonal ChernSimons partition function defined in Section 4. Expand the free energy
Then the reformulated invariants are defined by
We conjecture that
Conjecture 1.1 (Orthogonal LMOV).
and
Conjecture 1.2 (Degree).
where , is the valuation of the variable and is the sum of the lengths of the partition corresponding to each component of the link .
This conjecture is a mathematical formulation of the conjecture made by BouchardFloreaMariño [7], and the integer coefficients on the right hand side of the above conjecture is closely related to BPS numbers in string theory [7]. More recent progress can be found in [34], which is a refined version of [7]. The framing version can be found in [6, 38]. Our formulation is still quite different from that in [7, 34]. The reason for this is that [7, 34] uses representations of Hecke algebra, whereas our approach is based on representations of the BirmanMurakamiWenzl algebra, and uses typeB Schur function instead of typeA Schur function as the basis in the orthogonal ChernSimons partition function.
Theorems that partly answer the orthogonal LMOV conjecture proposed in this paper are listed below. For more precise statements of these theorems, see Sections 5, 7, 8 and 9.
Theorem 1.3.
The conjecture is true for all partitions when the link is trivial (namely is a disjoint union of unlinked unknots).
Theorem 1.4.
The conjecture is true for partitions of the shape , where for .
Theorem 1.5.
The conjecture is true if and only if it is true for partitions of the shape .
Theorem 1.6.
The conjecture asymptotically holds (for all partitions and all knots/links) as tends to .
Theorem 1.7.
The conjecture is true when is: the torus knots/links , where is odd/even, and each component of the partition is of the form , or ; the two components torus link for partition ; the three components torus link for the partition . These examples give evidence for the conjecture at nontrivial roots of unity.
We also prove the degree conjecture.
Theorem 1.8.
The following degree estimation holds
In addition, we use the cabling technique developed in [29] to calculate colored Kauffman polynomials for torus knots and links, which are employed to test the orthogonal LMOV conjecture (Theorem 1.7).
This paper is organized as follows: In Section 2, we review some basic knowledge of partitions, the BirmanMurakamiWenzl algebra and irreducible representation of the Brauer algebra. In Section 3, we review the definition of the quantum group invariants of links and use the cabling formula to simplify the computation of these invariants. As an application of the cabling formula, we obtain colored Kauffman polynomials of all torus knots and links for all partitions (irreducible representations). In Section 4, we define the ChernSimons partition function for orthogonal quantum groups and the corresponding reformulated invariants. Also, we compute the orthogonal ChernSimons partition function for disjoint union of unknots (Theorem 1.3). In Section 5, we propose a new orthogonal LMOV conjecture and degree conjecture. Then we test torus knots and links as supporting examples (Theorem 1.7), which can not be treated as special cases of the proof in the following sections. In Section 6, we obtain formulas of LickorishMillett type by using skein relations at the intersections of two different link components. This trick is also widely used in Section 7. Anyway, this section is quite independent and such LickorishMillett type formulas can also be treated as an application of the orthogonal LMOV conjecture, which is the starting point of this paper. In Section 7, we prove the equivalence between the vanishing of the first three coefficients of for trivial partitions (each component of partitions have only one box), predicted by the degree conjecture, and the LichorishMillett type formulas obtained in Section 6. We also prove the orthogonal LMOV conjecture for columnlike Young diagram (Theorem 1.4) as a generalization of such LichorishMillett type formulas. In Section 8 and 9, we prove that if the orthogonal LMOV conjecture is valid for the case of rows, then the orthogonal LMOV is valid for all partitions (Theorem 1.5). In addition, the proof of the degree conjecture is also presented there (Theorem 1.7), which implies that the orthogonal LMOV Conjecture asymptotically holds (for all partitions and all knots/links) as tends to (Theorem 1.6). In Section 10 (Appendix), we first compute explicit expressions of the ChernSimon partition function for the unknot. We then review an alternative definition of the colored Kauffman polynomial via the Markov trace (skein approach) and test the Hopf link for the orthogonal LMOV conjecture by using this new definition. We also give an explicit computation of the quantum trace for orthogonal quantum groups directly from the universal matrix. Finally, we list the character table of Brauer algebra and typeB Schur functions, whose specialization gives colored Kauffman polynomials of the unknot (quantum dimensions) for small partitions. These tables are mainly used to compute colored Kauffman polynomial for torus knots and links. The tables of the integers coefficients predicted by the orthogonal LMOV conjecture are also presented.
1.3. Acknowledgments
The authors greatly benefited from Nicolai Reshetikhin’s early participation in this project, and owe him a lot for his help, advices and support. We also thank Kefeng Liu, Pan Peng and Hao Zheng for explaining their works to us and many very useful discussions. We thank Marcos Mariño for communicating with us on this subject and his interest and numerous useful comments. We thank Francis Bonahon for his enthusiasm, advices, and support. Part of this work was done while the authors visited the Center of Mathematical Science at Zhejiang University. The second author also thanks the Hausdorff Institute of Mathematics at Bonn and IHÉS for their hospitality. The first draft of this paper was ready in the Fall 2008. The first author passed away in a tragic accident in 2009. The current version is presented here by the second author in memory of his good friend and collaborator Lin Chen.
2. Young Diagram and BirmanMurakamiWenzl Algebra
2.1. Partition and Young Diagram
A composition of , denoted by , is a finite sequence of positive integers such that
The number of parts in is called the length of and denote by . The size of composition is defined by
A partition is a composition such that
Denote by the set of all partitions. We identify a partition with its Young diagram.
If , we say is a partition of , denote by .
We use to denote the number of times that appears in . Denote the automorphism group of the partition by .
The order of is given by
There is another way to rewrite a partition in the following format
For Instance, we have
Define the following numbers associated to a partition .
2.2. Partitionable set and infinite series
Following the notations of [32], we present some basic knowledge of partitionable set here.
The concept of partition can be generalized to the following partitionable set.
A countable set is called a partitionable set if the following holds

is totally ordered.

is an Abelian semigroup with summation .

The minimum element in is the zeroelement of the semigroup, i.e., for any ,
For simplicity, we may briefly write instead of .
The following sets are examples of partitionable set:

The set of all nonnegative integers ;

The set of all partitions . The order of can be defined as follows:
, , iff , or and there exists a such that for and . The summation is taken to be and the zeroelement is .

. The order of is defined similarly as :
, , iff , or and there is a such that for and .
Define
is the zeroelement. Then is a partitionable set.
Let be a partitionable set. One can define partition with respect to in the similar manner as that of : a finite sequence of nonincreasing nonminimum elements in . We will call it an Spartition, the zero partition. Denote by the set of all partitions.
For an partition , we can define the automorphism group of in a similar way as that in the definition of traditional partition. Given , denote by the number of times that occurs in the parts of , we then have
Introduce the following quantities associated with ,
The following Lemma will be used in Section 4 to deduce the reformulated invariants.
Proof.
Note that
∎
2.3. BirmanMurakamiWenzl Algebra
The centralizer algebra
(2.1) 
for the standard representation of on is isomorphic, when , to the BirmanMurakamiWenzl algebra .
Let be the field of rational functions with two variables. For each positive integer , the BirmanMurakamiWenzl algebra is defined to be an algebra over as follows. The algebra is of one dimensional, and thus is identified to . For , is generated over by the generators and the relations
(A1) for
(A2) if
(A3)
(A4)
(A5) .
The first two properties are the braiding relations. The following two properties are immediate from the above definition
(P1) for
(P2) .
When the variable approaches to , while is fixing, the above BMW algebra specializes to the Brauer algebra , which is semisimple and isomorphic to the centralizer algebra if , cf. [8] and also [53]. The BirmanMurakamiWenzl algebras are semisimple except possibly when takes the value of roots of unity or for some integer . Obviousely, the BMW algebra is the deformation of the Brauer algebra.
2.4. Irreducible Representations of Brauer Algebras
For our purpose, we focus the generic case when the BMW Algebras are semisimple. In this situation, its irreducible representation can be described similar to the Brauer Algebras .
As the centralizer algebra , contains the group algebra as a direct summand, thus all the irreducible representations of are also irreducible representations of , labeled by partitions of the integer . Indeed, the set of irreducible representations of are bijective to the set of partitions of the integers , where [41, 51]. Thus the semisimple algebra can be decomposed into the direct sum of simple algebras
The work of Beliakova and Blanchet [4] constructed an explicit basis of the above decomposition. An up and down tableau is a tube of Young diagrams such that and each is obtained by adding or removing one box from . Let be a partition of . Denote by if , and we say an up and down tableau is of shape . There is a minimal path idempotent associated to each . Then the minimal central idempotent of correspond to the irreducible representation labeled by is given by
In particular, the dimension of the irreducible representations is the number of up and down tableau of shape . More detail can be found in [4, 51].
The characters table and the orthogonal relations can be found in [40, 41, 42]. The values of a character of is completely determined by its values on the set of elements , where is the conjugacy class of and is the conjugacy class in labeled by the partition of . The notion stands for the tangle in the following diagram.
where is a diagram in the conjugacy class of labeled by a partition of .
Denote the character of the irreducible representation of labeled by a partition for some , and denote by the character of the irreducible representation of labeled by a partition . It is known that when is a partition of , then for all and partition , and for partition coincide with the characters of the permutation group [41].
2.5. SchurWeyl Duality
Both and acts on the tensor product and their actions commute each other. As a bimodule, has the following decomposition
where runs through all the partitions of , (resp. ) is the irreducible representation of (resp. ) labeled by . A similar decomposition holds for the pair and .
A power symmetric function of a sequence of variables is defined by
For a partition ,
Denote the set of all the characters of . For each partition , we use to denote the typeB Schur function associated to with infinitely many variables , which are completely determined by the system of equations inductively
(2.2) 
The parameter is the structure constant in the definition of the Brauer algebra . The typeB Schur functions is independent of this parameter , as one can see from the character formula of Brauer algebra, given by A. Ram in [41] Theorem 5.1. If is a partition of , then is a symmetric polynomial of degree (not necessarily homogeneous).
Throughout this paper, we fix the following notations for partition set , where is the number of components of link .
For , denote
(2.3) 
and define
(2.4) 
Write
(2.5) 
for the sum of the length of each partition.
We denote , where .
Let denotes the set , then for the character of labeled by , a partition of , and the conjugacy class of labeled by .
3. Colored Kauffman Polynomials and Cabling Formula
3.1. Colored Kauffman Polynomials (Orthogonal Quantum Groups Invariants)
Let be the braid group of strands which is generated by with following defining relations:
(3.1) 
Every link can be represented by the closure of some element in braid group . This kind of braid representation is not unique. We fix such a braid representation, then we define the quantum group invariants of link via this braid. Finally we will see such kind of definition is independent of the choice of the braid representation.
Let be a finite dimensional complex simple Lie algebra and be the corresponding quantized enveloping algebra.
The ribbon category structure associated with is given by the following data:

Associated to each pair of modules and , there is an isomorphism
such that
for modules , , .
Given , , one has the following naturality condition:

There exists an element , called the enhancement of , such that
for any , . Here is the halfsum of all positive roots of .
Moreover, for every with , , one has the quantum trace

For any module , the ribbon structure associated to satisfies
The ribbon structure also satisfies the following naturality condition
for any .
Let be a link with components , , represented by the closure of . We associate each an irreducible representation of quantized universal enveloping algebra labeled by highest weight . In the sense of [41], these irreducible representations can be labeled by partitions. By abuse of notations, we use ’s to denote those partitions. Let be integers such that if the th strand of belongs to the th component of .
Let , be two modules labeling two outgoing strands of the crossing, the braiding (resp. ) is assigned as in following figure.
The above assignment will give a representation of on module . Namely, for any generator ,
define
and
Therefore, any link will provide an isomorphism
The representation of braid group on factors through the BMW algebra by sending to . By abuse of notations, we still denote this via .
The quantum trace
defines framing dependent link invariant of link .
In order to eliminate the framing dependency, we make the following refinement [29]
where is the writhe number of in , i.e., the number of positive crossing minus the number of negative crossings.
The above quantity is invariant under the Markov moves, hence is an invariant of the underlying link .
Quantum group invariants of links can be defined over any complex simple Lie algebra . However, in this paper, we mainly consider the quantum group invariants of links defined over . More generally, one can also include the case for and ; however, we will not do so, since the quantum group invariants associated to these Lie algebras all give the colored Kauffman polynomials. To distinguish from the quantum group corresponding to spin group, we only consider those representations parameterized by the highest weights in the root lattice of the Lie group , instead of the spin group. These highest weights are, similar to the case of , partitions of length at most , i.e .
Let’s consider , the quantized universal enveloping algebra of orthogonal lie algebra , which is generated by together with the following defining relations:
where is the Cartan matrix of
and the Serre type relations
where and the number is defined as
When , the universal enveloping algebra reduces to the Lie algebra .
Drinfeld [12] defined the universal matrix of as
(3.2) 
where denotes the set of positive roots and the exponential is of the form
The ribbon category structure is defined by letting for the above universal matrix, and taking to be . The operator switches the two components, and denotes the element in the Cartan subalgebra corresponding to .
The positive roots of are given by for and , where has eigenvalue when acting on the matrix element
in the Cartan subalgebra. The sum of the positive roots is given by
and
Alternatively, we can write
The universal matrix acting on for the natural representation of on is given by Turaev [50]:
where is the matrix with
and
The ribbon structure is equal to for .
Define the orthogonal quantum group invariants such that
(3.3) 
Then we want to compute the identity . We will first introduce the representation theory of the BMW algebra.
From now on, we only restrict ourselves in the case when the BirmanMurakamiWenzl algebra is semisimple and is large. The representations of can be described in the same way as the Brauer algebra . The semisimplicity implies that the representation of admits a direct sum decomposition
The multiplicities are all positive integers. In particular, any irreducible representation of appear as a direct summand of for integer . By Schur lemma,
where is isomorphic to the matrix algebra, labelled by the characters of as the decomposition of .
A minimal idempotent satisfies and the action of on the subspace is an irreducible representation. Another description of is that, there exist exactly one such that the restriction of to is nonzero, and it’s a diagonalizable matrix with exactly one eigenvalue 1 and all others 0.
Let be an element in , and the normal (or say, nonquantum) trace of its component via the above isomorphism is denoted by . Since and all the idempotents are elements in , they are finite linear combinations of products of the generators ’s and ’s, which imply is, in general, a rational function of and .
It is not hard to get the following identity from the Turaev’s [50] construction of universal matrix (See Section 10 for detail):
where is the standard representation of on .
More generally, we have the following lemma obtained by Reshetikhin [43]
Lemma 3.1.
For each partition with , one has
where .
This result can be understand in the following way. First we have
In [29], they use a different normalization for universal matrices, thus they have
and also a different corresponding normalization for factoring through the Hecke algebra via