Universal Racah matrices and adjoint knot polynomials: Arborescent knots

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• 作者：A. Mironov^a ; ^b ; ^c ; ^d ; ^{mironov@lpi.ru" class="auth_mail" title="E-mail the corresponding author} ; ^{mironov@itep.ru" class="auth_mail" title="E-mail the corresponding author} ; A. Morozov^b ; ^c ; ^d ; ^{morozov@itep.ru" class="auth_mail" title="E-mail the corresponding author}
• 刊名：Physics Letters B
• 出版年：2016
• 出版时间：10 April 2016
• 年：2016
• 卷：755
• 期：Complete
• 页码：47-57
• 全文大小：405 K

文摘

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 (SU_N${\mathit{SU}}_N$) and Kauffman (SO_N${\mathit{SO}}_N$) polynomials. For E₈$E_8$ 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 R$\mathcal{R}$-matrix, and for dealing with the adjoint polynomials one has to extend it to the previously unknown 6×6$6\times 6$ 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.