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HOMFLY and superpolynomials for figure eight knot in all symmetric and antisymmetric representations. (English) Zbl 1397.57012
Summary: Explicit answer is given for the HOMFLY polynomial of the figure eight knot \(4_1\) 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 \({\mathfrak H}_R={\mathfrak H}^{|R|}_{[1]}\) 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 \({\mathcal A}\)-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.

MSC:
57M25 Knots and links in the \(3\)-sphere (MSC2010)
Software:
Knot Atlas
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References:
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