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Nilpotent slices, Hilbert schemes, and the Jones polynomial

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Ciprian Manolescu

Symplectic Geometry · Submitted 31 Oct 2004

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim This variety is the intersection of a semisimple orbit with a transverse slice at a nilpotent in the Lie algebra $sl_{2m}.$ We exhibit bijections between a set of generators for the Seidel-Smith cochain complex, the generators in Bigelow's picture of the Jones polynomial, and the generators of the Heegaard Floer cochain complex for the double branched cover.

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arXiv IDmath/0411015 (version 3)
Pith IDTXJPQJ5J, Pith's permanent record of this paper and its review
CitationCiprian Manolescu. "Nilpotent slices, Hilbert schemes, and the Jones polynomial." arXiv:math/0411015 (2004). https://pith.science/paper/TXJPQJ5J
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,033 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.06 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.04 · compares with prior results 0.79 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v3. Results and the main claim are underlined; .

Seidel and Smith have constructed an invariant of links as the Floer cohomology for two Lagrangians inside a complex affine variety Y. This variety is the intersection of a semisimple orbit with a transverse slice at a nilpotent in the Lie algebra $sl_{2m}.$ We exhibit bijections between a set of generators for the Seidel-Smith cochain complex, the generators in Bigelow's picture of the Jones polynomial, and the generators of the Heegaard Floer cochain complex for the double branched cover. This is done by presenting Y as an open subset of the Hilbert scheme of a Milnor fiber.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 Seidel and Smith have constructed an invariant of links as the Floer cohomology for two Lagrangians inside a complex affine variety Y. background 0.99 0.05 0.09 asserted 1.00
s1 This variety is the intersection of a semisimple orbit with a transverse slice at a nilpotent in the Lie algebra $sl_{2m}.$ We exhibit bijections between a set of generators for the Seidel-Smith cochain complex, the generators in Bigelow's picture of the Jones polynomial, and the generators of the Heegaard Floer cochain complex for the double branched cover. result 0.56 0.90 0.62 asserted 1.00
s2 This is done by presenting Y as an open subset of the Hilbert scheme of a Milnor fiber. method 0.98 0.85 0.09 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.99 · evidence from proof or derivation 0.98.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

Seidel and Smith have constructed an invariant of links as the Floer cohomology for two Lagrangians inside a complex affine variety Y. This variety is the intersection of a semisimple orbit with a transverse slice at a nilpotent in the Lie algebra $sl_{2m}.$ We exhibit bijections between a set of generators for the Seidel-Smith cochain complex, the generators in Bigelow's picture of the Jones polynomial, and the generators of the Heegaard Floer cochain complex for the double branched cover. This is done by presenting Y as an open subset of the Hilbert scheme of a Milnor fiber.

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7 arXiv papers cite this in 12 sentences

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Cites it as background 1 paper
  • In particular we want to mention Manolescu's interpretation [ref] of Seidel-Smith symplectic Khovanov homology [ref] in terms of Bigelow's model for the Jones polynomial [ref] , see also section [ref].

    Categorified sl(N) invariants of colored rational tangles · Introduction and statement of results · 0.80

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