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Title:
Refined Knot Invariants and Hilbert Schemes
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Abstract:
Motivated by the string theory, M. Aganagic and S. Shakirov proposed a “refinement” to quantum invariants of knots, which can be formulated mathematically for torus knots using Macdonald polynomials. In a recent paper with A. Negut we proved that these “refined invariants” can be obtained as equivariant Euler characteristics of certain sheaves on the Hilbert scheme of points on the plane. Using localization at fixed points, one can get explicit formulae for these invariants. In some special cases, these results match a certain bivariate deformation of Catalan numbers defined by A. Garsia and M. Haiman.
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