Generalized Seifert surfaces and signatures of colored links
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- by David Cimasoni and Vincent Florens PDF
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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.References
- M. F. Atiyah and I. M. Singer, The index of elliptic operators. III, Ann. of Math. (2) 87 (1968), 546–604. MR 236952, DOI 10.2307/1970717
- A. Casson, C. M. Gordon, Cobordism of classical knots in $S^3$, Printed notes. Orsay (1975).
- A. J. Casson and C. McA. Gordon, On slice knots in dimension three, Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976) Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I., 1978, pp. 39–53. MR 520521
- David Cimasoni, A geometric construction of the Conway potential function, Comment. Math. Helv. 79 (2004), no. 1, 124–146. MR 2031702, DOI 10.1007/s00014-003-0777-6
- J. H. Conway, An enumeration of knots and links, and some of their algebraic properties, Computational Problems in Abstract Algebra (Proc. Conf., Oxford, 1967) Pergamon, Oxford, 1970, pp. 329–358. MR 0258014
- D. Cooper, Signatures of surfaces with applications to knot and link cobordism, Ph. D. thesis, University of Warwick, 1982.
- D. Cooper, The universal abelian cover of a link, Low-dimensional topology (Bangor, 1979) London Math. Soc. Lecture Note Ser., vol. 48, Cambridge Univ. Press, Cambridge-New York, 1982, pp. 51–66. MR 662427
- R. H. Crowell and D. Strauss, On the elementary ideals of link modules, Trans. Amer. Math. Soc. 142 (1969), 93–109. MR 247625, DOI 10.1090/S0002-9947-1969-0247625-1
- V. Florens, Signatures of colored links with application to real algebraic curves, J. Knot Theory Ramifications 14 (2005), no. 7, 883–918. MR 2187604, DOI 10.1142/S0218216505004093
- Vincent Florens and Patrick M. Gilmer, On the slice genus of links, Algebr. Geom. Topol. 3 (2003), 905–920. MR 2012958, DOI 10.2140/agt.2003.3.905
- R. H. Fox, Some problems in knot theory, Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961) Prentice-Hall, Englewood Cliffs, N.J., 1962, pp. 168–176. MR 0140100
- Patrick M. Gilmer, Configurations of surfaces in $4$-manifolds, Trans. Amer. Math. Soc. 264 (1981), no. 2, 353–380. MR 603768, DOI 10.1090/S0002-9947-1981-0603768-7
- Patrick M. Gilmer, On the slice genus of knots, Invent. Math. 66 (1982), no. 2, 191–197. MR 656619, DOI 10.1007/BF01389390
- Patrick Gilmer and Charles Livingston, Discriminants of Casson-Gordon invariants, Math. Proc. Cambridge Philos. Soc. 112 (1992), no. 1, 127–139. MR 1162937, DOI 10.1017/S0305004100070808
- C. McA. Gordon, On the $G$-signature theorem in dimension four, À la recherche de la topologie perdue, Progr. Math., vol. 62, Birkhäuser Boston, Boston, MA, 1986, pp. 159–180. MR 900251
- C. McA. Gordon, R. A. Litherland, and K. Murasugi, Signatures of covering links, Canadian J. Math. 33 (1981), no. 2, 381–394. MR 617628, DOI 10.4153/CJM-1981-032-3
- Richard Hartley, The Conway potential function for links, Comment. Math. Helv. 58 (1983), no. 3, 365–378. MR 727708, DOI 10.1007/BF02564642
- Louis H. Kauffman and Laurence R. Taylor, Signature of links, Trans. Amer. Math. Soc. 216 (1976), 351–365. MR 388373, DOI 10.1090/S0002-9947-1976-0388373-0
- J. Levine, Knot cobordism groups in codimension two, Comment. Math. Helv. 44 (1969), 229–244. MR 246314, DOI 10.1007/BF02564525
- J. P. Levine, Signature invariants of homology bordism with applications to links, Knots 90 (Osaka, 1990) de Gruyter, Berlin, 1992, pp. 395–406. MR 1177436
- J. P. Levine, Link invariants via the eta invariant, Comment. Math. Helv. 69 (1994), no. 1, 82–119. MR 1259607, DOI 10.1007/BF02564475
- A. Libgober, On the homology of finite abelian coverings, Topology Appl. 43 (1992), no. 2, 157–166. MR 1152316, DOI 10.1016/0166-8641(92)90137-O
- Jun Murakami, On local relations to determine the multi-variable Alexander polynomial of colored links, Knots 90 (Osaka, 1990) de Gruyter, Berlin, 1992, pp. 455–464. MR 1177442
- Kunio Murasugi, On a certain numerical invariant of link types, Trans. Amer. Math. Soc. 117 (1965), 387–422. MR 171275, DOI 10.1090/S0002-9947-1965-0171275-5
- S. Yu. Orevkov, Link theory and oval arrangements of real algebraic curves, Topology 38 (1999), no. 4, 779–810. MR 1679799, DOI 10.1016/S0040-9383(98)00021-4
- Stepan Yu. Orevkov, Plane real algebraic curves of odd degree with a deep nest, J. Knot Theory Ramifications 14 (2005), no. 4, 497–522. MR 2150745, DOI 10.1142/S0218216505003920
- S. Orevkov, in preparation.
- V. A. Rohlin, Two-dimensional submanifolds of four-dimensional manifolds, Funkcional. Anal. i Priložen. 5 (1971), no. 1, 48–60 (Russian). MR 0298684
- Makoto Sakuma, Homology of abelian coverings of links and spatial graphs, Canad. J. Math. 47 (1995), no. 1, 201–224. MR 1319696, DOI 10.4153/CJM-1995-010-2
- H. Seifert, Über das Geschlecht von Knoten, Math. Ann. 110 (1935), no. 1, 571–592 (German). MR 1512955, DOI 10.1007/BF01448044
- Lawrence Smolinsky, A generalization of the Levine-Tristram link invariant, Trans. Amer. Math. Soc. 315 (1989), no. 1, 205–217. MR 931532, DOI 10.1090/S0002-9947-1989-0931532-5
- A. G. Tristram, Some cobordism invariants for links, Proc. Cambridge Philos. Soc. 66 (1969), 251–264. MR 248854, DOI 10.1017/s0305004100044947
- H. F. Trotter, Homology of group systems with applications to knot theory, Ann. of Math. (2) 76 (1962), 464–498. MR 143201, DOI 10.2307/1970369
- O. Ja. Viro, Branched coverings of manifolds with boundary, and invariants of links. I, Izv. Akad. Nauk SSSR Ser. Mat. 37 (1973), 1241–1258 (Russian). MR 0370605
Additional Information
- David Cimasoni
- Affiliation: Department of Mathematics, University of California Berkeley, 970 Evans Hall, Berkeley, California 94720
- MR Author ID: 677173
- 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
- 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. - © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 360 (2008), 1223-1264
- MSC (2000): Primary 57M25
- DOI: https://doi.org/10.1090/S0002-9947-07-04176-1
- MathSciNet review: 2357695