Universal Racah matrices and adjoint knot polynomials
By now it is well established that the quantum dimensions of descendants of the adjoint representation can be described in a universal form, independent of a particular family of simple Lie algebras. The Rosso-Jones formula then implies a universal description of the adjoint knot polynomials for torus knots, which in particular unifies the HOMFLY () and Kauffman () polynomials. For the adjoint representation is also fundamental. We suggest to extend the universality from the dimensions to the Racah matrices and this immediately produces a unified description of the adjoint knot polynomials for all arborescent (double-fat) knots, including twist, 2-bridge and pretzel. Technically we develop together the universality and the ”eigenvalue conjecture”, which expresses the Racah and mixing matrices through the eigenvalues of the quantum -matrix, and for dealing with the adjoint polynomials one has to extend it to the previously unknown case. The adjoint polynomials do not distinguish between mutants and therefore are not very efficient in knot theory, however, universal polynomials in higher representations can probably be better in this respect.
Lebedev Physics Institute, Moscow 119991, Russia
ITEP, Moscow 117218, Russia
Institute for Information Transmission Problems, Moscow 127994, Russia
National Research Nuclear University MEPhI, Moscow 115409, Russia
Knot theory - is intimately connected to representation theory via the Reshetikhin-Turaev (RT) formalism -. For the simplest (torus) knots the Rosso-Jones formula - expresses knot invariants through just quantum dimensions and Casimir eigenvalues. For evaluating knot polynomials of other knots, one has to know the Racah matrices. In fact, for a broad family of arborescent (double-fat) knots [2, 23, 24] it is sufficient to know only two Racah matrices [11, 13, 14]. Moreover, in the case of self-contragredient representation they are proportional to each other so that one needs only one Racah matrix, and this is exactly the case of adjoint representation which we consider in this paper. However, in general also needed are the ”mixing matrices” [8, 17], which are contractions of several Racah so that to construct them one has to know a series of Racah matrices.
A remarkable discovery in representation theory is universality -, a possibility to describe quantities for all the simple Lie algebras by a universal formula, which is the same for unitary, orthogonal, symplectic and exceptional groups. It turns out that such universal description exists, provided one considers only adjoint representation and its descendants (representations, appearing in tensor powers of adjoint), instead of arbitrary descendants of the fundamental one.
Historically, the term ”universality” refers to the notion of the ”Universal Lie algebra”, introduced by Vogel in , which, roughly speaking, was intended to describe the -algebra of triple-ended Feynman diagrams (closely related to Connes-Kreimer description [40, 41]). These diagrams are related to the Vassiliev invariants and, on the physical side, to the perturbative expansion in Chern-Simons theory, and the universality came just as an observation, as it was also the case earlier . We call some quantity in the theory of (quantum) simple Lie algebras universal, if it can be expressed as a smooth symmetric function of three parameters , and takes values for a given simple Lie algebra at the corresponding points of Vogel’s Table, see (15) below in this text (for the ADE case it appeared already in ).
Recently in  it was suggested to extend the notion of universality to the adjoint knot polynomials, i.e. to express them in terms of three variables instead of the more conventional two . Conventional colored polynomials (in the ”-sector” of representation theory) appear on particular -dimensional slices of the space. For instance, the (HOMFLY) and (Kauffman) polynomials are described by choosing and respectively. However, in terms of the knot polynomials acquire an additional property: they are symmetric functions of these variables. In  such universal formulas were explicitly presented for a variety of knots that included 2-and 3-strand torus knots and links and the figure eight knot, for the 2-strand case they were later reproduced in . Actually, for the torus case this is not a big surprise, because the Rosso-Jones formula providing the knot polynomials in this case does not contain anything but quantum dimensions and Casimir eigenvalues, which are known to possess universal description [27, 29, 30, 31, 33]. Still , the results of  were originally obtained without any use of the Rosso-Jones formula, moreover, in the figure eight case this formula is unapplicable at all. The actual message of , which we make explicit in the present and the sequel paper , is that the universality can be lifted to the Racah and mixing matrices, and thus all the knot polynomials in adjoint family can actually be represented in the universal form. In this paper we make this idea explicit for the entire family of arborescent knots (which can be presented by double-fat graphs), evaluation of their knot polynomials having been discussed in detail in .
We achieve this by developing another challenging conjecture: the eigenvalue conjecture (EC) of  (see also ) expressing the mixing matrices between -matrices acting on different pairs of adjacent strands in a braid through the eigenvalues of -matrices themselves (actually, those eigenvalues are with being value of the second Casimir operator).
Actually, the EC in  is formulated for the 3-strand closed braids, and then the mixing matrices are actually the Racah matrices. At the same time, once the Racah matrix is known, ideas of  can be used to evaluate the colored polynomials for the arborescent knots, which is a family very different from the 3-strand one (sometimes wider, sometimes narrower). In fact, the 3-strand family per se can also be studied by the method of the present paper, but this is a separate story that will be reported elsewhere . Technically EC provides a solution to the Yang-Baxter equation
with diagonal in the form of orthogonal matrix ,
with all entries explicitly expressed through the eigenvalues ’s (the number of Yang-Baxter equations is exactly equal to the number of independent angles in the orthogonal matrix). In  such solution was explicitly provided for , for the purposes of the present paper we need an extension to , what is not at all straightforward (as emphasized also in ).
Note that in terms of representation theory, is constructed as the Racah matrix that relates the two maps: and (for the sake of simplicity, we discuss here only the case of knots, not links when one is not obliged to consider three coinciding representations ). This matrix is involved in evaluating the knot invariants from 3-strand braid representation. At the same time, evaluating the arborescent knot invariants involves the Racah matrices and that relate accordingly the maps with and with , where denotes the representation contragredient to . In the case of adjoint representation , which is self-contragredient, all three matrices , and are equal to each other and the property of one of them are immediately inherited by the remaining ones. In particular, since this case is multiplicity free, all three matrices are symmetric (due to symmetricity of ), and their first row is (the property inherited from )
where denotes the quantum dimension of the representation .
What we do in the present paper, we solve two problems at once: we use the coincidence of these Racah matrices in order to apply the EC to , additionally using its symmetricity and the form of the first row. This allows us to restore the matrix and then we use to construct the universal adjoint polynomials of the arborescent knots.
The paper is organized as follows. In section 2 we construct a mixing matrix that is further used in section 3 for constructing the universal adjoint knot polynomials of various arborescent knots. Section 4 contains some comments of extension from the family of arborescent knots and also a discussion of properties of the polynomials obtained in the present paper. The concluding remarks are in section 5.
2 mixing matrix
In accordance with what is said in the Introduction, we need to construct the Racah matrix that maps and . Since
is decomposed into six irreps (they are actually irreducible with respect to a Lie algebra, multiplied by a discrete symmetry of Dynkin diagram), the relevant Racah matrix is , as it was already claimed. Now we are going to apply the EC.
To understand the structure of eigenvalue formulas, it deserves looking at the case. Generic symmetric orthogonal matrix –solution to (2) in this case is just
with . Substitution into (1) gives
This structure is inherited by the mixing matrices for higher . It involves a parameter and different expressions for diagonal and the squares of the off-diagonal elements of , with simple denominators and some -dependent polynomials of and in the numerators.
In particular, the mixing matrix for can be obtained using the coincidence of matrices , and as was explained in the Introduction, which adds to the properties of mixing matrix also the properties of and . The result reads:
squares of the off-diagonal elements:
and where and are the complements of in the set .
2.2 Tensor square of adjoint representation
As we already wrote, the tensor square of adjoint representation is given by formula (4). The last two representations and belong to the antisymmetric square , while the first four to the symmetric square . The corresponding -matrix eigenvalues, which are equal to with being the second Casimir, are
The quantum dimensions that makes a content of the first row of the mixing matrix, (3) are
and the quantum dimensions for and are obtained from by cyclic permutations of the triple . These representations for the concrete case of are
where we extended the group by automorphisms of its Dynkin diagram, so that the sum of two last representations becomes one irreducible representation of the extended group. For these representations are
Dimensions, eigenvalues, Racah/mixing matrices and knot polynomials for particular Lie algebras arise under substitutions of , , from Vogel’s table
where all exceptional simple Lie algebras belong to the line at special values of parameter:
2.3 The universal Racah matrix for adjoint
Now are finally ready to write down the mixing matrix for the concrete eigenvalues of the adjoint (9) which we further use for constructing the universal knot polynomials for the arborescent knots. In the latter case we use the standard notation for the Racah matrix , hence the notation below.
The matrix is symmetric and independent items are ():
In practical calculations one needs also matrix elements like
which are (partly) listed in .
3 Universal adjoint knot polynomials
3.1 Mixing matrix and torus knots: checks of consistency
By construction, the matrix (17) possesses all the necessary properties: it is symmetric, orthogonal (thus, its square is equal to 1), satisfies the Yang-Baxter equation (1). Here we check that it gives correct contributions to the universal adjoint polynomials of the 2- and 3-strand torus knots and links earlier obtained in . In the 2-strand case these are given by
in full accord with .
In the 3-strand case we can either use the universal Rosso-Jones formula from  or derive it from the mixing matrices:
In other words, the coefficients , which depend on only modulo , can be extracted either from the Adams rule or evaluated from traces over the intertwiner spaces in at the r.h.s. To do this, we need to know the eigenvalues of , where eigenvalues of are quantum Casimirs of representations from , leading to , and mixing matrices are made out of them by the eigenvalue conjecture of .
comes from , the corresponding is one dimensional and
For the space is 6-dimensional, the corresponding mixing matrix is exactly (17), and
For the space is 3-dimensional, and
and so on for other representations .
In result neither nor contributes to the universal adjoint polynomials of the 3-strand torus knots, when , but both contribute in the case of links, when , in full accordance with the answers in :
plus cyclic permutations of the triple for the and cases. Like it is done in , denotes a sum of three dimensions, equal to
with . The three individual dimensions in this case are quite sophisticated even in the classical limit of , but all the three have identical Casimirs and enter the formulas for knots as a sum, which has a long, but explicit expression through .
3.2 Obtaining arborescent knots
Using the Racah matrix (17) and the corresponding eigenvalues (9), one can immediately evaluate the universal adjoint polynomials of the arborescent knots. Their expressions through this data can be found, e.g., in  or even extracted from the tables in . The final answers for the universal adjoint polynomials are quite long, hence, we do not list them here, but in order to illustrate how the procedure works we collected in  the universal adjoint polynomials obtained in this way: for all arborescent knots with no more than 8 crossings (i.e. all knots with no more than 8 crossings except for single in the Rolfsen table, ), and also for many arborescent knots with 9 and 10 crossings. Note that among these examples there are numerous knots that can be presented as well as 3,4,5,6-strand closed braids on one side, and knots that can be presented by double-fat graphs with up to 6 fingers, plenty of them being non-pretzel and even non-starfish ones on the other side: the list is quite representative. However, to make the text readable, we discuss below the universal adjoint polynomials for just two simple families of arborescent knots: for the twist and pretzel knots.
Before coming to these families, we make an important comment about the framing factor. That is, despite there is no difference between the Racah matrices , and for the adjoint representation, and between the sets of the corresponding eigenvalues (i.e. the universal adjoint knot polynomial does not depend on orientation, since the adjoint representation is contragredient), this is literally true only in group theory. For knots the situation is a little trickier. The point is that the group theory objects like universal -matrix correspond to the so called ”vertical” framing in knot theory, which is different from the ”topological” one. In result, the topologically invariant knot polynomials, which we are looking for, keep some memory about the difference between and , because their transformations from vertical to topological framings are different. This was the origin of additional factors in knot polynomials of the previous subsection (e.g. each -matrix in the previous subsection has to be multiplied by in order to obtain the correct formulas (26), (3.1) from (23)-(25), and an additional factor in (19)) and this will produce some extra common powers of in our formulas below, they are the only memory of orientation dependence which survives in the universal sector of knot theory. We just insert these powers of wherever necessary without going into lengthy comments. In practice, say, for the pretzel knots, one picks up a representation from the tables of  or , like and inserts a factor per each non-overlined item, while nothing for the overlined one, i.e. a total of for this realization of or for pretzel representation of , or for pretzel representation of , see formulas (37) and (38) below.
3.3 Twist knots
The twist knots are described by the knot diagrams like (here is equal to 3 and the knot is ):
The general formula for the universal adjoint polynomial for the twist knot is
where denotes the diagonal matrix with elements .
It can be represented in the general evolution formula  for the universal adjoint polynomial for this family of knots
with and the coefficients