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Generalized Seifert surfaces and signatures of colored links


Authors: David Cimasoni and Vincent Florens
Journal: Trans. Amer. Math. Soc. 360 (2008), 1223-1264
MSC (2000): Primary 57M25
DOI: https://doi.org/10.1090/S0002-9947-07-04176-1
Published electronically: October 23, 2007
MathSciNet review: 2357695
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Abstract: In this paper, we use `generalized Seifert surfaces' to extend the Levine-Tristram signature to colored links in $ S^3$. This yields an integral valued function on the $ \mu$-dimensional torus, where $ \mu$ is the number of colors of the link. The case $ \mu=1$ corresponds to the Levine-Tristram signature. We show that many remarkable properties of the latter invariant extend to this $ \mu$-variable generalization: it vanishes for achiral colored links, it is `piecewise continuous', and the places of the jumps are determined by the Alexander invariants of the colored link. Using a $ 4$-dimensional interpretation and the Atiyah-Singer $ G$-signature theorem, we also prove that this signature is invariant by colored concordance, and that it provides a lower bound for the `slice genus' of the colored link.


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  • 1. M. Atiyah, I. Singer, The index of elliptic operators. III, Ann. of Math. 87 (1968), 546-604. MR 0236952 (38:5245)
  • 2. A. Casson, C. M. Gordon, Cobordism of classical knots in $ S^3$, Printed notes. Orsay (1975).
  • 3. A. Casson, C. M. Gordon, On slice knots in dimension three, Proc. Symp. in Pure Math. XXX (1978), Part 2, 39-53. MR 0520521 (81g:57003)
  • 4. D. Cimasoni, A geometric construction of the Conway potential function, Comment. Math. Helv. 79 (2004), no. 1, 124-146. MR 2031702 (2005d:57005)
  • 5. J. Conway, An enumeration of knots and links, and some of their algebraic properties, In Computational Problems in Abstract Algebra (Oxford, 1967), Proc. Conf. (Pergamon Press, Oxford, 1967), pp. 329-358. MR 0258014 (41:2661)
  • 6. D. Cooper, Signatures of surfaces with applications to knot and link cobordism, Ph. D. thesis, University of Warwick, 1982.
  • 7. D. Cooper, The universal abelian cover of a link, Low-dimensional topology (Bangor, 1979), pp. 51-66, London Math. Soc. Lecture Note Ser., 48, Cambridge Univ. Press, Cambridge-New York, 1982. MR 0662427 (83g:57002)
  • 8. R. Crowell, D. Strauss, On the elementary ideals of link modules, Trans. Amer. Math. Soc. 142 (1969), 93-109. MR 0247625 (40:889)
  • 9. V. Florens, Signatures of colored links with application to real algebraic curves, J. Knot Theory Ramifications 14 (2005), 883-918. MR 2187604
  • 10. V. Florens, P. Gilmer, On the slice genus of links, Algebr. Geom. Topol. (2003), 905-920. MR 2012958 (2004h:57006)
  • 11. R. Fox, Some problems in knot theory, 1962 Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961) 168-176 Prentice-Hall, Englewood Cliffs, N.J.MR 0140100 (25:3523)
  • 12. P. Gilmer, Configurations of Surfaces in 4-manifolds, Trans. Amer. Math. Soc. 264 (1981), 353-380. MR 0603768 (83h:57027)
  • 13. P. Gilmer, On the Slice Genus of Knots, Invent. Math. 66 (1982), 191-197.MR 0656619 (83g:57003)
  • 14. P. Gilmer, C. Livingston, Discriminant of Casson-Gordon invariants, Math. Proc. Cambridge Philos. Soc. 112 (1992), no. 1, 127-139.MR 1162937 (94e:57007)
  • 15. C. M. Gordon, On the $ G$-Signature theorem in dimension 4, Proceedings Okahoma Topology Conference 1978. MR 0900251
  • 16. C. M. Gordon, R. Litherland , K. Murasugi, Signatures of covering links, Canad. J. Math. 33 (1981), no. 2, 381-394. MR 0617628 (83a:57006)
  • 17. R. Hartley, The Conway potential function for links, Comment. Math. Helv. 58 (1983), no. 3, 365-378. MR 0727708 (85h:57006)
  • 18. L. Kauffman, L. Taylor, Signature of links, Trans. Amer. Math. Soc. 216 (1976), 351-365. MR 0388373 (52:9210)
  • 19. J. Levine, Knot cobordism groups in codimension two, Comment. Math. Helv. 44 (1969), 229-244. MR 0246314 (39:7618)
  • 20. J. Levine, Signature invariants of homology bordism with applications to links, Knots 90 (Osaka, 1990), 395-406, de Gruyter, Berlin, 1992.MR 1177436 (94g:57026)
  • 21. J. Levine, Link invariants via the eta invariant, Comment. Math. Helv. 69 (1994), no. 1, 82-119. MR 1259607 (95a:57009)
  • 22. A. Libgober, On the homology of finite abelian coverings, Topology Appl. 43 (1992), no. 2, 157-166. MR 1152316 (93e:57003)
  • 23. J. Murakami, On local relations to determine the multi-variable Alexander polynomial of colored links, Knots 90 (Osaka, 1990), 455-464, de Gruyter, Berlin, 1992. MR 1177442 (93k:57020)
  • 24. K. Murasugi, On a certain numerical invariant of link types, Trans. Amer. Math. Soc. 117 (1965), 387-422. MR 0171275 (30:1506)
  • 25. S. Orevkov, Link theory and oval arrangements of real algebraic curves, Topology 38 (1999), no. 4, 779-810. MR 1679799 (2000b:14066)
  • 26. S. Orevkov, Plane real algebraic curves of odd degree with a deep nest, J. Knot Theory Ramifications 14 (2005), 497-522. MR 2150745 (2006d:14068)
  • 27. S. Orevkov, in preparation.
  • 28. D. Rohlin, Two-dimensional submanifolds of four-dimensional manifolds, Funkcional. Anal. i Prilozen. 5 (1971), no. 1, 48-60.MR 0298684 (45:7733)
  • 29. M. Sakuma, Homology of abelian coverings of links and spacial graphs, Can. J. Math. 47 (1995), no.1, 201-224. MR 1319696 (96d:57008)
  • 30. H. Seifert, Über das Geschlecht von Knoten, Mathematische Annalen 110 (1934), 571-592. MR 1512955
  • 31. L. Smolinsky, A generalization of the Levine-Tristram link invariant, Trans. Amer. Math. Soc. 315 (1989), no. 1, 205-217.MR 0931532 (89m:57021)
  • 32. A. G. Tristram, Some cobordism invariants for links, Proc. Camb. Philos. Soc. 66 (1969), 251-264. MR 0248854 (40:2104)
  • 33. H. Trotter, Homology of group systems with applications to knot theory, Ann. of Math. 76 (1962), no. 2, 464-498. MR 0143201 (26:761)
  • 34. O. Viro, Branched coverings of manifolds with boundary and link invariants I, Math. USSR Izvestia 7 (1973), 1239-1256. MR 0370605 (51:6832)

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Additional Information

David Cimasoni
Affiliation: Department of Mathematics, University of California Berkeley, 970 Evans Hall, Berkeley, California 94720
Email: cimasoni@math.berkeley.edu

Vincent Florens
Affiliation: Departamento Ãlgebra, Geometrã y Topologã, Universidad de Valladolid, Prado de la Magdalena s/n, 47011 Valladolid, Spain
Address at time of publication: Section de Mathématiques, Université de Genève, 2-4 rue du Lièvre, Case Postale 64, 1211 Genève 4, Switzerland
Email: vincent_florens@yahoo.fr, vincent.florens@math.unige.ch

DOI: https://doi.org/10.1090/S0002-9947-07-04176-1
Keywords: Colored link, Seifert surface, Levine-Tristram signature, slice genus.
Received by editor(s): May 6, 2005
Received by editor(s) in revised form: August 23, 2005
Published electronically: October 23, 2007
Additional Notes: The first author was supported by the Swiss National Science Foundation.
The second author was supported by Marie-Curie, MCHF-2001-0615.
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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