# 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

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topBorodzik, 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

top- V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko, Singularities of differentiable maps. Volume 2, (2012), Birkhäuser/Springer, New York Zbl1290.58001MR2919697
- M. Borodzik, Morse theory for plane algebraic curves, J. of Topology 5 (2012), 341-365 Zbl1251.57004MR2928080
- M. Borodzik, A. Némethi, Spectrum of plane curves via knot theory, J. London Math. Soc. 86 (2012), 87-110 Zbl1247.32027MR2959296
- G. Burden, H. Zieschang, Knots, 2ed, (2003), Walter de Gruyter & co., Berlin Zbl1009.57003MR1959408
- J. C. Cha, C. Livingston, KnotInfo: Table of Knot Invariants
- Louis H. Kauffman, On knots, 115 (1987), Princeton University Press, Princeton, NJ Zbl0627.57002MR907872
- A. Kawauchi, A survey on knot theory, (1996), Birkhäuser–Verlag, Basel, Boston, Berlin Zbl0861.57001MR1417494
- Patrick W. Keef, On the $S$-equivalence of some general sets of matrices, Rocky Mountain J. Math. 13 (1983), 541-551 Zbl0533.57011MR715777
- Dmitry Kerner, András Némethi, The Milnor fibre signature is not semi-continuous, Topology of algebraic varieties and singularities 538 (2011), 369-376, Amer. Math. Soc., Providence, RI Zbl1225.32030MR2777830
- R. A. Litherland, Signatures of iterated torus knots, Topology of low-dimensional manifolds (Proc. Second Sussex Conf., Chelwood Gate, 1977) 722 (1979), 71-84, Springer, Berlin Zbl0412.57002MR547456
- Charles Livingston, Knot theory, 24 (1993), Mathematical Association of America, Washington, DC Zbl0887.57008MR1253070
- John Milnor, Singular points of complex hypersurfaces, (1968), Princeton University Press, Princeton, N.J. Zbl0184.48405MR239612
- John Milnor, On isometries of inner product spaces, Invent. Math. 8 (1969), 83-97 Zbl0177.05204MR249519
- Kunio Murasugi, On a certain numerical invariant of link types, Trans. Amer. Math. Soc. 117 (1965), 387-422 Zbl0137.17903MR171275
- Kunio Murasugi, Knot theory & its applications, (2008), Birkhäuser Boston Inc., Boston, MA Zbl1138.57001MR2347576
- Yasutaka Nakanishi, A note on unknotting number, Math. Sem. Notes Kobe Univ. 9 (1981), 99-108 Zbl0481.57002MR634000
- András Némethi, The real Seifert form and the spectral pairs of isolated hypersurface singularities, Compositio Math. 98 (1995), 23-41 Zbl0851.14015MR1353284
- András Némethi, Variation structures: results and open problems, Singularities and differential equations (Warsaw, 1993) 33 (1996), 245-257, Polish Acad. Sci., Warsaw Zbl0855.32019MR1449162
- Walter D. Neumann, Invariants of plane curve singularities, Knots, braids and singularities (Plans-sur-Bex, 1982) 31 (1983), 223-232, Enseignement Math., Geneva Zbl0586.14023MR728588
- J. H. M. Steenbrink, Mixed Hodge structure on the vanishing cohomology, Real and complex singularities (Proc. Ninth Nordic Summer School/NAVF Sympos. Math., Oslo, 1976) (1977), 525-563, Sijthoff and Noordhoff, Alphen aan den Rijn Zbl0373.14007MR485870
- J. H. M. Steenbrink, Semicontinuity of the singularity spectrum, Invent. Math. 79 (1985), 557-565 Zbl0568.14021MR782235
- A. N. Varchenko, On the semicontinuity of the spectra and estimates from above of the number of singular points of a projective hypersurface, Doklady Akad. Nauk. 270 (1983), 1294-1297 Zbl0537.14003MR712934
- Henryk Żołądek, The monodromy group, 67 (2006), Birkhäuser Verlag, Basel Zbl1103.32015MR2216496

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