Hodge–type structures as link invariants
Annales de l'Institut Fourier, Volume 63 (2013) no. 1, pp. 269-301.

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.

En se fondant sur des analogies avec la théorie de Hodge des singularités isolées des hypersurfaces, nous construisons des invariants numériques de type de Hodge pour un entrelacs quelconque, pas forcément algébrique, dans une sphère de dimension trois. Nous appelons ces invariants les H-nombres. Ils contiennent la même information sur les entrelacs, que la partie non-dégénérée de la matrice de Seifert modulo S-équivalence réelle. Nous étudions leurs propriétés, en particulier, donnons une formule explicite pour les signatures de Tristram et Levine et les polynômes d’Alexander de haut ordre en termes des H-nombres.

De plus, motivés par la théorie des singularités, nous introduisons le spectre d’un entrelacs, qui, lui aussi, peut être exprimé en termes des H-nombres. Nous établissons quelques propriétés de semicontinuité.

Ces propriétés, peuvent être reliées aux relations skein. Cependant, elles ne sont pas aussi précises que les relations skein classiques.

DOI: 10.5802/aif.2761
Classification: 57M25, 32S25, 14D07, 14H20
Keywords: Seifert matrix, Hodge numbers, Alexander polynomial, Tristram–Levine signature, variation structure, semicontinuity of the spectrum
Mot clés : La matrice de Seifert, les nombres de Hodge, le polynôme d’Alexander, les signatures de Tristram et Levine, la structure de variation, la semicontinuité du spectre
Borodzik, Maciej 1; Némethi, András 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)
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     title = {Hodge{\textendash}type structures as link invariants},
     journal = {Annales de l'Institut Fourier},
     pages = {269--301},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {63},
     number = {1},
     year = {2013},
     doi = {10.5802/aif.2761},
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     language = {en},
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PB  - Association des Annales de l’institut Fourier
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Borodzik, Maciej; Némethi, András. Hodge–type structures as link invariants. Annales de l'Institut Fourier, Volume 63 (2013) no. 1, pp. 269-301. doi : 10.5802/aif.2761. https://aif.centre-mersenne.org/articles/10.5802/aif.2761/

[1] Arnold, V. I.; Gusein-Zade, S. M.; Varchenko, A. N. Singularities of differentiable maps. Volume 2, Modern Birkhäuser Classics, Birkhäuser/Springer, New York, 2012 (Monodromy and asymptotics of integrals, Translated from the Russian by Hugh Porteous and revised by the authors and James Montaldi, Reprint of the 1988 translation) | MR | Zbl

[2] Borodzik, M. Morse theory for plane algebraic curves, J. of Topology, Volume 5 (2012) no. 2, pp. 341-365 | DOI | MR | Zbl

[3] Borodzik, M.; Némethi, A. Spectrum of plane curves via knot theory, J. London Math. Soc., Volume 86 (2012) no. 1, pp. 87-110 | DOI | MR | Zbl

[4] Burden, G.; Zieschang, H. Knots, 2ed, Walter de Gruyter & co., Berlin, 2003 | MR | Zbl

[5] Cha, J. C.; Livingston, C. KnotInfo: Table of Knot Invariants (http://www.indiana.edu/ knotinfo)

[6] Kauffman, Louis H. On knots, Annals of Mathematics Studies, 115, Princeton University Press, Princeton, NJ, 1987 | MR | Zbl

[7] Kawauchi, A. A survey on knot theory, Birkhäuser–Verlag, Basel, Boston, Berlin, 1996 | MR | Zbl

[8] Keef, Patrick W. On the S-equivalence of some general sets of matrices, Rocky Mountain J. Math., Volume 13 (1983) no. 3, pp. 541-551 | DOI | MR | Zbl

[9] Kerner, Dmitry; Némethi, András The Milnor fibre signature is not semi-continuous, Topology of algebraic varieties and singularities (Contemp. Math.), Volume 538, Amer. Math. Soc., Providence, RI, 2011, pp. 369-376 | DOI | MR | Zbl

[10] Litherland, R. A. Signatures of iterated torus knots, Topology of low-dimensional manifolds (Proc. Second Sussex Conf., Chelwood Gate, 1977) (Lecture Notes in Math.), Volume 722, Springer, Berlin, 1979, pp. 71-84 | MR | Zbl

[11] Livingston, Charles Knot theory, Carus Mathematical Monographs, 24, Mathematical Association of America, Washington, DC, 1993 | MR | Zbl

[12] Milnor, John Singular points of complex hypersurfaces, Annals of Mathematics Studies, No. 61, Princeton University Press, Princeton, N.J., 1968 | MR | Zbl

[13] Milnor, John On isometries of inner product spaces, Invent. Math., Volume 8 (1969), pp. 83-97 | DOI | MR | Zbl

[14] Murasugi, Kunio On a certain numerical invariant of link types, Trans. Amer. Math. Soc., Volume 117 (1965), pp. 387-422 | DOI | MR | Zbl

[15] Murasugi, Kunio Knot theory & its applications, Modern Birkhäuser Classics, Birkhäuser Boston Inc., Boston, MA, 2008 (Translated from the 1993 Japanese original by Bohdan Kurpita, Reprint of the 1996 translation [MR1391727]) | DOI | MR | Zbl

[16] Nakanishi, Yasutaka A note on unknotting number, Math. Sem. Notes Kobe Univ., Volume 9 (1981) no. 1, pp. 99-108 | MR | Zbl

[17] Némethi, András The real Seifert form and the spectral pairs of isolated hypersurface singularities, Compositio Math., Volume 98 (1995) no. 1, pp. 23-41 | Numdam | MR | Zbl

[18] Némethi, András Variation structures: results and open problems, Singularities and differential equations (Warsaw, 1993) (Banach Center Publ.), Volume 33, Polish Acad. Sci., Warsaw, 1996, pp. 245-257 | MR | Zbl

[19] Neumann, Walter D. Invariants of plane curve singularities, Knots, braids and singularities (Plans-sur-Bex, 1982) (Monogr. Enseign. Math.), Volume 31, Enseignement Math., Geneva, 1983, pp. 223-232 | MR | Zbl

[20] Steenbrink, J. H. M. Mixed Hodge structure on the vanishing cohomology, Real and complex singularities (Proc. Ninth Nordic Summer School/NAVF Sympos. Math., Oslo, 1976), Sijthoff and Noordhoff, Alphen aan den Rijn, 1977, pp. 525-563 | MR | Zbl

[21] Steenbrink, J. H. M. Semicontinuity of the singularity spectrum, Invent. Math., Volume 79 (1985) no. 3, pp. 557-565 | DOI | MR | Zbl

[22] Varchenko, A. N. On the semicontinuity of the spectra and estimates from above of the number of singular points of a projective hypersurface, Doklady Akad. Nauk., Volume 270 (1983) no. 6, pp. 1294-1297 | MR | Zbl

[23] Żołądek, Henryk The monodromy group, Instytut Matematyczny Polskiej Akademii Nauk. Monografie Matematyczne (New Series) [Mathematics Institute of the Polish Academy of Sciences. Mathematical Monographs (New Series)], 67, Birkhäuser Verlag, Basel, 2006 | MR | Zbl

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