HOMFLY and superpolynomials for figure eight knot in all symmetric and antisymmetric representations

Abstract

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 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.

Link polynomial calculus and the AENV conjecture

“…We present the results in the form of the differential expansion of [36], generalizing the dream-like formulas of [25] for the figure eight knot. The series L 2k+1 in representation [r] ⊗ [1] .…”

“…The answers are represented either in the differential hierarchy (Z-expansion) form of [25,36], which is convenient to control the representation dependence, or in the evolution based form of [29,43], convenient to control the dependence on the shape of the knot. These two representations look very different, but in every particular case one can easily convert between them.…”

“…After getting these explicit formulas, we converted them into a differential expansion form a la [25,36]. As usual, such formulas in symmetric representations have a pronounced q-hypergeometric form (in accordance with a generic statement of [45]) and are easily continued to arbitrary values of r, s and t.…”

“…In the paper, we also apply the evolution method of [25,43] to the family of 2-component 3-strand links (see figure 2), which includes the Hopf link, the Whitehead link, L7a6, L9a36, L11a360 etc in accordance with the classification of [47]. In this way, we obtain the HOMFLY polynomials for this whole family in the case of the both symmetric representations, the result reads:…”

Checks of integrality properties in topological strings

“…This formalism is successfully developed in [31] and [33] and has already allowed us to find the inclusive Racah matrices for R = [2,2] and even R = [3,1]. In combination with the differential expansion method [142][143][144][145][146][147][148][149][150], this provides extensions to other rectangular representations. Further progress (for other nonrectangular representations) is expected after developing the ∆-technique briefly outlined in [33].…”