Duke Mathematical Journal

The Hilbert scheme of a plane curve singularity and the HOMFLY polynomial of its link

Alexei Oblomkov and Vivek Shende

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The intersection of a complex plane curve with a small three-sphere surrounding one of its singularities is a nontrivial link. The refined punctual Hilbert schemes of the singularity parameterize subschemes supported at the singular point of fixed length and whose defining ideals have a fixed number of generators. We conjecture that the generating function of Euler characteristics of refined punctual Hilbert schemes is the HOMFLY polynomial of the link. The conjecture is verified for irreducible singularities yk=xn whose links are the (k,n) torus knots, and for the singularity y4=x7x6+4x5y+2x3y2 whose link is the (2,13) cable of the trefoil.

Article information

Duke Math. J., Volume 161, Number 7 (2012), 1277-1303.

First available in Project Euclid: 4 May 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14H20: Singularities, local rings [See also 13Hxx, 14B05]
Secondary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}


Oblomkov, Alexei; Shende, Vivek. The Hilbert scheme of a plane curve singularity and the HOMFLY polynomial of its link. Duke Math. J. 161 (2012), no. 7, 1277--1303. doi:10.1215/00127094-1593281. https://projecteuclid.org/euclid.dmj/1336142076

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