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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)
##### Keywords:
Chern-Simons theories; topological strings; quantum groups
Knot Atlas
Full Text:
##### References:
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