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Link Polynomial Calculus and AENV Conjecture
by
Semeon Arthamonov
Rutgers, The State University of New Jersey
Coauthors: A. Mironov, A. Morozov, And. Morozov
Recently Aganagic, Ekholm, Ng, and Vafa conjectured a relation between the augmentation variety in the large N limit of the colored HOMFLY and quantum A-polynomials. In this talk I will describe the methods used for direct confirmation of this conjecture for certain links including Whitehead link and Borromean rings.
It appears that colored knot polynomials possess an internal structure (we call it Z-expansion) which behaves naturally under inclusion of the representation into the product of the fundamental ones. In particular, for some large families of links the colored HOMFLY polynomial for symmetric and anti-symmetric representations can be presented as a truncated sum of a certain q-hypergeometric series. The latter allows us to extend the formulas for the arbitrary symmetric representations and study the asymptotic of the colored HOMFLY polynomials for large symmetric representations.
In addition I will say a few words about the extension of Z-expansion beyond the symmetric representations for some simplest examples. Although for generic representation we no longer have those cute truncated q-hypergeometric series we still have some interesting structure beyond the HOMFLY and superpolynomials. In particular, the introduction of the recently developed fourth grading in all existing examples can be presented as an elegant redefinition of the constituents of Z-expansion.
Date received: December 8, 2014
Copyright © 2014 by the author(s). The author(s) of this work and the organizers of the conference have granted their consent to include this abstract in Topology Atlas. Document # cbjz-42.