Congruent Skein Relations for Colored HOMFLY-PT invariants and Colored Jones Polynomials
Colored HOMFLY-PT invariant, the generalization of the colored Jones polynomial, is one of the most important quantum invariants of links. This paper is devoted to investigating the basic structures of the colored HOMFLY-PT invariants of links. By using the HOMFLY-PT skein theory, firstly, we show that the (reformulated) colored HOMFLY-PT invariants actually lie in the ring . Secondly, we establish some symmetric formulas for colored HOMFLY-PT invariants of links, which include the rank-level duality as an easy consequence. Finally, motivated by the Labastida-Mariño-Ooguri-Vafa conjecture for framed links, we propose congruent skein relations for (reformulated) colored HOMFLY-PT invariants which are the generalizations of the skein relation for classical HOMFLY-PT polynomials. Then we study the congruent skein relation for colored Jones polynomials. In fact, we obtain a succinct formula for the case of knot. As an application, we prove a vanishing result for Reshetikhin-Turaev invariants of a family of 3-manifolds. Finally we study the congruent skein relations for quantum invariants.
Key words and phrases:Colored HOMFLY-PT invariants, Integrality, Symmetries, Rank-level duality, LMOV questions, Skein relations
1991 Mathematics Subject Classification:Primary 57M25, Secondary 57M27 81R50
- 1 Introduction
- 2 Colored HOMFLY-PT invariants
- 3 Integrality property
- 4 Symmetries, Level-rank duality
- 5 More results on reformulated colored HOMFLY-PT invariants
- 6 Congruent skein relations for (reformulated) colored HOMFLY-PT invariants
- 7 Congruent skein relations for colored Jones polynomials
- 8 Appendix: examples for congruent skein relations
The HOMFLY-PT polynomial is a two variables link invariant which was first discovered by Freyd-Yetter, Lickorish-Millet, Ocneanu, Hoste and Przytychi-Traczyk. In , Jones constructed the HOMFLY-PT polynomial by studying the representation of Hecke algebra. Let be an oriented link in , the HOMFLY-PT polynomial satisfies the following skein relation,
with the initial value , we will use the notation to denote the unknot throughout this paper. We denote by the Conway triple of an oriented link.
One can calculate the HOMFLY-PT polynomial for any given oriented link recursively through the above formula (1.1). Based on the work  of Turaev, the HOMFLY-PT polynomial can be obtained from the quantum invariant associated with the fundamental representation of the quantum group by letting . More generally, if we consider the quantum invariants associated with arbitrary irreducible representations of , by letting , we get the colored HOMFLY-PT invariants . See  for detailed definition of the colored HOMFLY-PT invariants through quantum group invariants of . The colored HOMFLY-PT invariants have an equivalent definition through the satellite invariants in HOMFLY-PT skein theory which, we refer to [1, 14] for a nice explanation of this equivalence. By using this approach, for a link with -components , . Let , where and denotes the set of all the partitions of positive integers. The colored HOMFLY-PT invariant of colored by is given by
where denotes the HOMFLY-PT polynomial of the link decorated by the element , where each is in the skein of annulus . For two partitions and , we let , where is the value of the character of symmetric group at the conjugate class . From the view of HOMFLY-PT skein theory, the element takes a simple form and has nice properties(see Section 2 for the detailed descriptions of the skein elements and ). So it is natural to study the following reformulated colored HOMFLY-PT invariants which are given by
In the first part of this paper, we obtain an integrality theorem for the (reformulated) colored HOMFLY-PT invariants by applying the HOMFLY-PT skein theory. By definition, a priori the colored HOMFLY-PT invariants lie in the ring with the elements admitted as denominators for . However, we can show that the reformulated colored HOMFLY-PT invariants actually belong to the subring , where we use notation throughout this paper. More precisely, we have
For any link with components, and ,
In the HOMFLY-PT skein theory, the two elements and satisfy the relation So we have the close relationship between the colored HOMFLY-PT invariants and reformulated colored HOMFLY-PT invariants . As the applications of Theorem 1.1, we establish the following symmetric properties:
Given a link with components, and , we have
These symmetries in Theorem 1.2 are very general. For example, combing (1.5) and (1.7), we obtain the symmetry:
which is referred as the rank-level duality in [21, 16]. Moreover, for a knot , if we use to denote the mirror of , then we have (See formula (4.19) in ). By formula (1.5), it is straightforward to obtain
which is just the formula (5) in .
1.3. Congruent skein relations
The seminal work  of E. Witten showed that Chern-Simons gauge theory provides a natural way to study the quantum invariants. In this framework, the expectation value of Wilson loop along a link in gives a topological invariant of the link depending on the representation of the gauge group. N. Reshetikhin and V. Turaev  gave a mathematical construction of this link invariant by using the representation theory of the quantum group. In particular, the gauge group with irreducible representation will give rise to the colored HOMFLY-PT invariant of the link . In another fundamental work of Witten , the Chern-Simons gauge theory on a three-manifold was interpreted as an open topological string theory on with topological branes wrapping inside . Furthermore, Gopakumar-Vafa  conjectured that the large limit of Chern-Simons gauge theory on is equivalent to the closed topological string theory on the resolved conifold. This highly nontrivial string duality was first checked for the case of the unknot by Ooguri-Vafa . Later, a series of work [21, 20] based on the large Chern-Simons/topological string duality, conjectured an expansion of the Chern-Simons partition functions in terms of an infinite sequence of integer invariants, which is called the Labastida-Mariño-Ooguri-Vafa (LMOV) conjecture. This integrality conjecture serves as an essential evidence of the Chern-Simons/topological string duality and was proved in . When considering the framing dependence for Chern-Simons gauge theory, the integrality structure is even more amazing as described in . In , two authors K. Liu and P. Peng paved a new way to study this framing dependence integrality structure conjecture (we call it framed LMOV conjecture in the following). In this framework, the framed LMOV conjecture provides us the interesting congruent skein relation for the reformulated colored HOMFLY-PT invariant .
In particular, when with row partitions , for . We use the notation to denote the reformulated colored HOMFLY-PT invariant for simplicity. By this definition, for a link with -components, is equal to the classical (framing dependence) HOMFLY-PT polynomial by multiplying a factor , i.e . The skein relation for the classical HOMFLY-PT polynomial leads to the skein relation for as follow:
when the crossing is the self-crossing of a component of the link , and
when the crossing is the linking of two different components of the link .
We use the notation to denote the link obtained by adding a positive kink to one of the component of link . By the relation (2.5),
By using the framing change formulas showed in Section 5, we establish the follow formula for :
When is a prime,
We see that has nice properties and the similar behaviors with . A natural question is if there exists the similar skein relation for ? Motivated by the new approach to the framed LMOV conjecture , we propose the following congruent skein relation for the reformulated colored HOMFLY-PT invariant as follow:
For any link and a prime number , we have
when the crossing is the self-crossing of a knot, and
when the crossing is the linking of two different components of the link . Where the notation denotes And , .
The complete proof of the above two relations is unknown, but some partial results will be given in Section 6 and appendix. For examples, we have
Let be the knot obtained by adding a positive kink to knot , and be the knot obtained by adding a negative kink to knot , let , then forms a Conway triple and the congruent skein relation (1.14) holds.
For the Conway triple , =unknot with two negative kinks and with one positive kink. The congruent skein relation (1.14) holds for .
Consider the torus link . Let , for the Conway triple , , , the congruent skein relation (1.14) holds for .
Similarly, for the Conway triple , , , the congruent skein relation (1.15) holds for .
1.4. Colored Jones polynomials
Colored Jones polynomial can be viewed as the special case of the colored HOMFLY-PT invariant:
So it is natural to consider if there exists the similar congruent skein relation for colored Jones polynomials . Please note that we use a little different symbol for the colored Jones polynomial in this paper, here denotes the Sym ( is the fundamental representation), which is the -dimension irreducible representation of . We prove the following congruent skein relation for any knot .
For any positive integers and ,
In fact, by using the cyclotomic expansion formula for colored Jones polynomial due to K. Habiro , we prove the following equivalent result.
For any knot , and positive integers ,
In particular, taking , we obtain
For any knot , ,
As an important application of Theorem 1.10, we obtain a vanishing result of the Reshetikhin-Turaev invariant for certain dimensional oriented closed manifolds. Let us denote by the -manifold obtained from by doing a -surgery () along a knot . Let , be the -dimension colored Jones polynomial of discussed above at the roots of unity . According to [30, 34], for odd integer , the Reshetikhin-Turaev invariants of can be calculated by the following formula:
where denotes certain function depending only on .
Inspired by the numerical phenomenon observed in , we prove the following
Theorem 1.12 (Vanishing of Reshetikhin-Turaev invariants).
For odd , if () and odd , then we have the vanishing of Reshetikhin-Turaev invariants as follows
This vanishing result surprises us a bit. After the communications with E. Witten , he told us that there should be a physical interpretation behind this phenomenon .
We also propose the following congruent skein relations for the quantum invariant for .
For a knot , for any positive integer and , we have
Finally these congruent skein relations for quantum invariants leads to Volume Conjectures for quantum invariants recently studied in .
The rest of this paper is organized as follows. In Section 2, we
introduce the HOMFLY-PT skein model to give the definition of
(reformulated) colored HOMFLY-PT invariants. In Section 3, we prove
the integrality theorem for the reformulated colored HOMFLY-PT
invariants. In Section 4, we establish the symmetries for colored
HOMFLY-PT invariants, including the rank-level duality as
applications. In Section 5, We provide more results on the
(reformulated) colored HOMFLY-PT invariants such as framing changing
formulas, which could be seen as the preparation for our study of
the congruent skein relations. In Section 6, we first propose a
conjecture of congruent skein relation for reformulated colored
HOMFLY-PT invariants, then we prove them in some special cases.
Furthermore, we show an application of the congruent skein
relations. Colored Jones polynomial can be regarded as the special
case of the colored HOMFLY-PT invariant. In Section 7, we prove a
congruent skein relation for colored Jones polynomial by using the
cyclotomic expansion of the colored Jones polynomial of knot. Then
we show a vanishing result of the Reshetikhin-Turaev invariants for
certain 3-manifols as an interesting application of this congruent
skein relation for colored Jones polynomials. Furthermore, we
conjectured link case of congruent skein relations for colored Jones
polynomials and knot case of congruent skein relations for
quantum invariants. In the appendix, we provide some sample examples
to illustrate the congruent skein relations for reformulated colored
HOMFLY-PT invariants, colored Jones polynomials and
Acknowledgements. We would like to thank R. Kashaev, J. Murakami, N. Reshetikhin, E. Witten and Tian Yang for valuable discussions with us. Q. Chen thank Dror Bar-Natan and Scott Morrison for communicating on KnotTheory, Package of Mathematica. and Q. Chen also thank CMS at Zhejiang University for their hospitality. Both Q. Chen and S. Zhu thank Shanghai Center for Mathematical Science for their hospitality. The research of S. Zhu is supported by the National Science Foundation of China grant No. 11201417 and the China Postdoctoral Science special Foundation No. 2013T60583.
2. Colored HOMFLY-PT invariants
2.1. Partitions and symmetric functions
A partition is a finite sequence of positive integers such that . The length of is the total number of parts in and denoted by . The weight of is defined by . If , we say is a partition of and denoted as . The automorphism group of , denoted by Aut(), contains all the permutations that permute parts of by keeping it as a partition. Obviously, Aut() has the order where denotes the number of times that occurs in .
Every partition is identified to a Young diagram. The Young diagram of is a graph with boxes on the -th row for , where we have enumerated the rows from top to bottom and the columns from left to right. Given a partition , we define the conjugate partition whose Young diagram is the transposed Young diagram of : the number of boxes on -th column of equals to the number of boxes on -th row of , for .
The following numbers associated with a given partition are used frequently in this article:
Obviously, is an even number and .
In the following, we will use the notation to denote the set of all the partitions of positive integers. Let be the partition of , i.e. the empty partition. Define , and the tuple of .
The power sum symmetric function of infinite variables is defined by Given a partition , define The Schur function is determined by the Frobenius formula
where is the character of the irreducible representation of the symmetric group corresponding to , we have if . The orthogonality of character formula gives
2.2. HOMFLY-PT skein theory
We follow the notations in . Define the coefficient ring with the elements admitted as denominators for . Let be a planar surface, the framed HOMFLY-PT skein of is the -linear combination of the orientated tangles in , modulo the two local relations as showed in Figure 1 where ,
It is easy to follow that the removal an unknot is equivalent to time a scalar , i.e we have the relation showed in Figure 2.
2.2.1. The plane
When , it is easy to follow that every element in can be represented as a scalar in . For a link with a diagram , the resulting scalar is the (framed unreduced) HOMFLY-PT polynomial of the link . I.e. . We use the convention for the empty diagram, so . The two relations showed in Figure 1 lead to
The classical HOMFLY-PT polynomial of a link is given by
where denotes the writhe number of link .
2.2.2. The rectangle
When is a rectangle with inputs at the top and outputs at the bottom. Let be the skein of -tangles. See Figure 3 for an element in .
Composing -tangles by placing one above another induces a product which makes into the Hecke algebra with the coefficients ring , where . has a presentation generated by the elementary brads subjects to the braid relations
and the quadratic relations .
2.2.3. The annulus
When is the annulus, we denote . is a commutative algebra with the product induced by placing the annulus one outside other. For any element , we use to denote the closure of as showed in Figure 4.
It is clear that . As an algebra, is freely generated by the set , for is the closure of the braid , and is the empty diagram . It follows that contains two subalgebras and which are generated by and . We denote the image of the closure map as . Thus . The linear subspace has a useful interpretation as the space of symmetric polynomials of degree in variables , for large enough . can be viewed as the algebra of the symmetric functions.
2.3. Basic elements in the skein of annulus
2.3.1. Turaev’s geometrical basis of
The element is the closure of the braid . Its mirror image is the closure of the braid . Given a partition of with length , we define the monomial . Then the monomials becomes a basis of which is called the Turaev’s geometric basis of .
Moreover, let be the closure of the braid . We define the element in as . There exist some explicit geometric relations between the elements , and .
2.3.2. Symmetric function basis of
The subalgebra can be interpreted as the ring of symmetric functions in infinite variables . The correspondence of the power sum symmetric function in is denoted by . Moreover, we have the identity
Denoted by the closures of idempotent elements in the Hecke algebra . It was showed by Lukac  that represent the Schur functions in the interpretation as symmetric functions. Hence forms a basis of . Furthermore, the Frobenius formula (2.2) gives
For brevity, the following notations will be used throughout the paper.
In particular, . For , we introduce
2.5. Definitions of the colored HOMFLY-PT invariants
Let be a framed link with components with a fixed numbering. For diagrams in the skein model of annulus with the positive oriented core , a link decorated with , denoted by , is constructed by replacing every annulus by the annulus with the diagram such that the orientations of the cores match. Each has a small backboard neighborhood in the annulus which makes the decorated link into a framed link (see Figure 5 for a framed trefoil decorated by skein element ).
In particular, when , where is the partition of a positive integer , for . The framed colored HOMFLY-PT invariant of decorated by is defined to be the framed HOMFLY-PT invariant of the decorated link , i.e. . By adding a framing factor to eliminate the framing dependency
The (framing-independence) colored HOMFLY-PT invariant is given by
The reformulated colored HOMFLY-PT invariants are defined as:
From the view of the HOMFLY-PT skein theory, the reformulated colored HOMFLY-PT invariants or are simpler than the colored HOMFLY-PT invariant , since the expression of is simpler than and has the nice property, see  for the similar statement. Therefore, it is natural to study the reformulated colored HOMFLY-PT invariant or instead of .
3. Integrality property
We have introduced in Section 2, is the correspondence of the symmetric power function in the skein . The geometric representation of is given by
where is the closure of the braid . Let
In particular, is the identity element in . Thus , for a link with -components. For convenience, in the following, we will also use the notation to denote and the notation to denote . By the skein relation (2.4) of , we have
where if the crossing is a self-crossing of a knot, if this crossing is a linking between two components of a link. Let be the unknot, we have . By using (3.3), it is obvious that
For any link , .
Furthermore, we have
For any link with components, and ,
We first consider the knot case, for a knot and a partition of length . We will show that .
We let be the -cabling of the knot which is a link of components. Then
As to the case of the link with components. Let , where . We let be the -cabling of the link which is a link of components. Similarly, we have
4. Symmetries, Level-rank duality
In this section, we will prove the symmetries for colored HOMFLY-PT invariants as showed in the introduction.
Given a link with components, and , we have the following symmetry: