HOMFLY and superpolynomials for figure eight knot in all symmetric and antisymmetric representationsстатья
Статья опубликована в высокорейтинговом журнале
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Дата последнего поиска статьи во внешних источниках: 14 сентября 2013 г.
Авторы:
Itoyama H.,
Mironov A.,
Morozov A.,
Morozov And
Аннотация:Explicit answer is given for the HOMFLY polynomial of the figure eight knot 41 in arbitrary symmetric representation R = [p]. It generalizes the old answers for p = 1 and 2 and the recently derived results for p = 3, 4, which are fully consistent with the Ooguri-Vafa conjecture. The answer can be considered as a quantization of the H_R=H^|R|_[1] identity for the “special” polynomials (they define the leading asymptotics of HOMFLY at q = 1), and arises in a form, convenient for comparison with the representation of the Jones polynomials as sums of dilogarithm ratios. In particular, we construct a difference equation (“non-commutative A -polynomial”) in the representation variable p. Simple symmetry transformation provides also a formula for arbitrary antisymmetric (fundamental) representation R = [1^p], which also passes some obvious checks. Also straightforward is a deformation from HOMFLY to superpolynomials. Further generalizations seem possible to arbitrary Young diagrams R, but these expressions are harder to test because of the lack of alternative results, even partial.