Eigenvalue conjecture and colored Alexander polynomials
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Abstract
We connect two important conjectures in the theory of knot polynomials. The first one is the property \(Al_R(q) = Al_{[1]}(q^{R})\) for all single hook Young diagrams R, which is known to hold for all knots. The second conjecture claims that all the mixing matrices \(U_{i}\) in the relation \(\mathcal{R}_i = U_i\mathcal{R}_1U_i^{1}\) between the ith and the first generators \(\mathcal{R}_i\) of the braid group are universally expressible through the eigenvalues of \(\mathcal{R}_1\). Since the above property of Alexander polynomials is very well tested, this relation provides new support to the eigenvalue conjecture, especially for \(i>2\), when its direct check by evaluation of the Racah matrices and their convolutions is technically difficult.
1 Introduction
An indisputable advantage of knot theory from the point of view of representation theory is that the former provides a set of quantities that adequately capture and reveal the basic hidden properties of the latter. These quantities, knot polynomials [1, 2, 3, 4, 5, 6, 7, 8] are the most natural in quantum field theory (QFT) realization of knot theory: they are just the Wilson loop averages in the topological Chern–Simons model [9, 10], which is one of the simplest in the family of Yang–Mills theories. The power of knot polynomial QFT methods in knot and representation theories is an impressive manifestation of the effectiveness of string theory approach to mathematical problems, especially when their calculational (algebraic) aspects are concerned.
 Realize the knot \(\mathcal{K}\) as a closure of an mstrand braid; a lot of such realizations is possible for any \(\mathcal{K}\), equivalence of the HOMFLY polynomials for different choices is guaranteed by invariance under the Reidemeister moves. Then (see [13, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40])where all the dependence on A is localized in the quantum dimensions (qgraded traces) \(\chi ^*\).$$\begin{aligned} H_R^\mathcal{K}(A,q) = \sum _{Q \vdash mR} C_{RQ}(q) \cdot \frac{\chi _Q^*}{\chi _R^*}(A,q) \end{aligned}$$(4)
 The quantum dimensions are given by the (hook) product formula over all boxes of the Young diagram:where \(\{x\}=xx^{1}\), while \(l,a,l',a'\) are the lengths of legs, arms, colegs and coarms, respectively.$$\begin{aligned} \chi _Q^* = \prod _{\square \in Q} \frac{\{Aq^{l_{_\square }'a_{_\square }'}\}}{\{q^{l_{_\square }+a_{_\square }+1}\}} \end{aligned}$$(5)
 If the knot \(\mathcal{K}\) is realized as a closure of the mstrand braid \(\Big (a_{1,1},\ldots ,a_{1,m1}a_{2,1},\ldots a_{2,m1} \ldots \Big )\), then the coefficients \(C_{RQ}\) in (4) are actually equal to [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40]$$\begin{aligned} C_{_{RQ}} = \mathrm{Tr}_{_{V_Q}} \Big ( \mathcal{R}_1^{a_{1,1}}\ldots \ \mathcal{R}_{m1}^{a_{1,m1}} \,\mathcal{R}_1^{a_{2,1}}\ldots \ \mathcal{R}_{m1}^{a_{2,m1}}\, \ldots \Big ).\nonumber \\ \end{aligned}$$(7)
This is why one can work with \(\mathcal{R}\), which acts already in the space of intertwining operators \(V^{(2)}_Y\). One can diagonalize, say, \(\mathcal{R}_1\) with acts on the first two strands. However, all matrices \(\mathcal{R}_i\) cannot be diagonalized at once. For \(m>2\) each multiplicity space \(V_Q^{(2)}\), arising in the mth tensor power of R, contains descendants of different Y from the second level, and while \(\mathcal{R}_1\) remains diagonal, the other \(\mathcal{R}_i\) (and thus elementary building blocks of \(U_i\) [35, 37]) after a proper ordering of columns and rows are blockdiagonal matrices with blocks of the size \(V_Q^{(2)}\otimes V_Q^{(2)}\) [35, 37, 38].
 It is a simple exercise to check that exactly the same is true for the ratios of quantum dimensions \(\chi _Q^*/\chi _R^*\):$$\begin{aligned} \left. \frac{\chi ^*_{[rmk,1^{ms+k}]} }{\chi ^*_{[r,1^s]}}\right _{A=1} = \frac{[r+s]}{[m(r+s)]} = \frac{\big [R\big ]}{\big [mR\big ]} \end{aligned}$$(11)

For \(m>2\) just the same reasoning would work with the only correction: the mixing matrices U emerge. For (2) to be true, it is sufficient if Umatrices depend on \(R=[r,1^s]\) through \(q^{r+s}=q^{R}\) only – and this is exactly what follows from the eigenvalue conjecture of [18]. The conjecture claims that the \(V_Q\otimes V_Q\) block of U, associated with representation Q, are made entirely from the normalized eigenvalues \(\epsilon _Y q^{\varkappa _Y}\) of the \(\mathcal{R}\)matrix \(\mathcal{R}_1\) for Y on the path from R to Q, and (10) shows that in our single hook case these are in turn made exactly from \(q^{r+s}=q^{R}\) (“normalized” means that the “shifts” in (10) can be neglected).
This is the main claim of the present letter. However, in practice this statement is evidence in favor of the eigenvalue conjecture rather than of (2). This is because (2) is very easy to check, once colored HOMFLY is known, and recent advances in HOMFLY calculus [41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53] provided quite a number of examples, which allows one to consider (2) rather well tested. The situation with the eigenvalue conjecture is much worse: it was quite difficult to check it for the matrix sizes (dimensions of W(Q)) 2,3,4,5 even in the simplest case of \(m=3\) strands. For size 6 it was validated very recently within the framework of the knot universality of [54, 55] and by application to advanced Racah calculus in [46, 48, 49]. This was an important step, because, beginning from the size 6, the eigenvalue conjecture does not immediately follow from the Yang–Baxter relations only [18, 56]; still for knot calculus it works well. Evidence for the eigenvalue hypotheses for a higher number of strands \(m>3\) is still nearly negligible. Mixing matrices are now not just Racah matrices but their convolutions [57, 58], which are extremely difficult to calculate, and not much has been yet done since [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40]. In this text we presented the claim that, if true for all m, the eigenvalue conjecture would explain the wellestablished equation (2), and this is new and reasonably strong evidence. It is far from proving anything, both because (2) is not proved and because the eigenvalue conjecture is sufficient, but not necessary for (2) to hold. Still, this new relation between the two conjectures should attract new attention to both of them and hopefully lead to a considerably better understanding.
From this interpretation of (2) it becomes clear what is so special about the single hook diagrams. For R with h hooks contributing to the sum (4) at \(A=1\) will also be the hhook diagrams Q, which will be parameterized by \(2h1\) parameters instead of a single k. Analysis of this situation is now possible and straightforward, however, it is not a surprise that the answer is more sophisticated than (2).
Notes
Acknowledgements
This work was performed at the Institute for Information Transmission Problems with the financial support of the Russian Science Foundation (Grant no. 145000150).
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