Hodge–type structures as link invariants

Maciej Borodzik[1]; András Némethi[2]

  • [1] University of Warsaw Institute of Mathematics ul. Banacha 2 02-097 Warsaw (Poland)
  • [2] A. Rényi Institute of Mathematics Reáltanoda u. 13-15 1053 Budapest(Hungary)

Annales de l’institut Fourier (2013)

  • Volume: 63, Issue: 1, page 269-301
  • ISSN: 0373-0956

Abstract

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Based on some analogies with the Hodge theory of isolated hypersurface singularities, we define Hodge–type numerical invariants of any, not necessarily algebraic, link in a three–sphere. We call them H–numbers. They contain the same amount of information as the (non degenerate part of the) real Seifert matrix. We study their basic properties, and we express the Tristram–Levine signatures and the higher order Alexander polynomial in terms of them. Motivated by singularity theory, we also introduce the spectrum of the link (determined from these H –numbers), and we establish some semicontinuity properties for it.These properties can be related with skein–type relations, although they are not so precise as the classical skein relations.

How to cite

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Borodzik, Maciej, and Némethi, András. "Hodge–type structures as link invariants." Annales de l’institut Fourier 63.1 (2013): 269-301. <http://eudml.org/doc/275547>.

@article{Borodzik2013,
abstract = {Based on some analogies with the Hodge theory of isolated hypersurface singularities, we define Hodge–type numerical invariants of any, not necessarily algebraic, link in a three–sphere. We call them H–numbers. They contain the same amount of information as the (non degenerate part of the) real Seifert matrix. We study their basic properties, and we express the Tristram–Levine signatures and the higher order Alexander polynomial in terms of them. Motivated by singularity theory, we also introduce the spectrum of the link (determined from these $H$–numbers), and we establish some semicontinuity properties for it.These properties can be related with skein–type relations, although they are not so precise as the classical skein relations.},
affiliation = {University of Warsaw Institute of Mathematics ul. Banacha 2 02-097 Warsaw (Poland); A. Rényi Institute of Mathematics Reáltanoda u. 13-15 1053 Budapest(Hungary)},
author = {Borodzik, Maciej, Némethi, András},
journal = {Annales de l’institut Fourier},
keywords = {Seifert matrix; Hodge numbers; Alexander polynomial; Tristram–Levine signature; variation structure; semicontinuity of the spectrum; Alexander polynomials; Nakanishi index; Tristram-Levine signature},
language = {eng},
number = {1},
pages = {269-301},
publisher = {Association des Annales de l’institut Fourier},
title = {Hodge–type structures as link invariants},
url = {http://eudml.org/doc/275547},
volume = {63},
year = {2013},
}

TY - JOUR
AU - Borodzik, Maciej
AU - Némethi, András
TI - Hodge–type structures as link invariants
JO - Annales de l’institut Fourier
PY - 2013
PB - Association des Annales de l’institut Fourier
VL - 63
IS - 1
SP - 269
EP - 301
AB - Based on some analogies with the Hodge theory of isolated hypersurface singularities, we define Hodge–type numerical invariants of any, not necessarily algebraic, link in a three–sphere. We call them H–numbers. They contain the same amount of information as the (non degenerate part of the) real Seifert matrix. We study their basic properties, and we express the Tristram–Levine signatures and the higher order Alexander polynomial in terms of them. Motivated by singularity theory, we also introduce the spectrum of the link (determined from these $H$–numbers), and we establish some semicontinuity properties for it.These properties can be related with skein–type relations, although they are not so precise as the classical skein relations.
LA - eng
KW - Seifert matrix; Hodge numbers; Alexander polynomial; Tristram–Levine signature; variation structure; semicontinuity of the spectrum; Alexander polynomials; Nakanishi index; Tristram-Levine signature
UR - http://eudml.org/doc/275547
ER -

References

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