Refined knot invariants and Hilbert schemes

Abstract

ABSTRACT. We consider the construction of refined Chern-Simons torus knot invariants by M. Aganagic and S. Shakirov from the DAHA viewpoint of I. Cherednik. We give a proof of Cherednik's conjecture on the stabilization of superpolynomials, and then use the results of O. Schiffmann and E. Vasserot to relate knot invariants to the Hilbert scheme of points on C 2 . Then we use the methods of the second author to compute these invariants explicitly in the uncolored case. We also propose a conjecture relating these constructions to the rational Cherednik algebra, as in the work of the first author, A. Oblomkov, J. Rasmussen and V. Shende. Among the combinatorial consequences of this work is a statement of the m n shuffle conjecture.

Words

“…Possible definitions of the superpolynomial for torus knots T (m,n) have recently been suggested by Angnanovic and Shakirov [AS11] (see also [AS12]), Cherednik [Che13], and Oblomkov, Rasmussen, and Shende [ORS12]. All three methods seem to give the same polynomial, and in fact Gorsky and Negut [GN13] have proved the descriptions in [AS11] and [Che13] do in fact give the same polynomial. The description in [AS11] is in terms of Macdonald polynomials, and Gorsky first realized that when m = n + 1, if we use the Cherednik parametrization, then the superpolynomial can be expressed as the function C n (q, t, −a) from (210), giving a completely new interpretation for the q, t-Schröder.…”

“…, x n ). Gorsky and Negut [GN13] show how the results of Aganagic and Shakirov on torus knot invariants can be expressed in terms of Macdonald polynomials using advanced objects such as the Hilbert scheme. Bergeron, Garsia, Leven, and Xin [BGLX14a], [BGLX14b] have shown how this Macdonald polynomial construction can be done combinatorially with plethystic symmetric function operators, and in fact they define operators Q (m,n) for any relatively prime (m, n) by a recursive procedure.…”

“…For example, Gorsky and Negut [GN13] prove that when t = 1/q the q, t-Catalan for (m, n), obtained by taking the coefficient of s 1 n in Q (m,n) (−1) n , reduces to…”

On universal knot polynomials

“…The Sp(N ) case can be obtained from the SO(N ) one by the substitution N −→ −N , transposition of Young diagrams and renormalization of scalar product in algebra (or, equivalently, on the language of Chern-Simons theory, by renormalization of coupling constant), see [9,91,92] for gauge theories' side of this equivalence, and [47,49] for that in knots theory. Some extra modifications are needed in the case of superpolynomials, see [47] and [79][80][81][82][83].…”

Процедура Каблирования Для Раскрашенных Полиномов ХОМФЛИ

“…Однако большинство приме-ров представляются полностью загадочными. Получение общих формул для серий узлов и представлений было бы крайне полезно не только с точки зрения изучения самих полиномов ХОМФЛИ, но и для исследования некоторых связанных с ними сюжетов таких, как разностные уравнения и τ -функции [71]- [77], [85], [95], [96], су-перполиномы [54]- [57], [61]- [70] и гомологии Хованова [109]- [115].…”