Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-28T13:22:20.448Z Has data issue: false hasContentIssue false
  • English
  • Français

Links not concordant to the Hopf link

Published online by Cambridge University Press:  24 February 2014

STEFAN FRIEDL  and
MARK POWELL
STEFAN FRIEDL
Affiliation:
Fakultät für Mathematik, Universität Regensburg, Germany. e-mail: [email protected]
MARK POWELL
Affiliation:
Department of Mathematics, Indiana University, Bloomington, Indiana, U.S.A. e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We give new Casson–Gordon style obstructions for a two–component link to be topologically concordant to the Hopf link.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 156 , Issue 3 , May 2014 , pp. 425 - 459
Copyright
Copyright © Cambridge Philosophical Society 2014 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[CF10]Cha, J. C. and Friedl, S. Twisted torsion invariants and link concordance. To appear in Forum Math., Preprint: http://arxiv.org/abs/1001.0926, (2010).Google Scholar
[CG86]Casson, A. and Gordon, C. McA.Cobordism of classical knots. In A la Recherche de la Topologie Perdue Progr. Math. vol. 62 (Birkhauser Boston, 1986), pages 181–199.Google Scholar
[Cha07]Cha, J. C.The structure of the rational concordance group of knots. Mem. Amer. Math. Soc. 189 (885) (2007). x+95.Google Scholar
[Cha10]Cha, J. C.Link concordance, homology cobordism, and Hirzebruch-type defects from iterated p-covers. J. Eur. Math. Soc. 12 (2010), 555–610.CrossRefGoogle Scholar
[Cha12]Cha, J. C. Symmetric Whitney tower cobordism for bordered 3–manifolds and links. arXiv:1204.4968 [math.GT] (2012).Google Scholar
[CK99a]Cha, J. C. and Ko, K. H.On equivariant slice knots. Proc. Amer. Math. Soc. 127 (7) (1999), 2175–2182.Google Scholar
[CK99b]Cha, J. C. and Ko, K. H.Signature invariants of links from irregular covers and non-abelian covers. Math. Proc. Camb. Philos. Soc. 127 (1) (1999), 67–81.CrossRefGoogle Scholar
[CK02]Cha, J. C. and Ko, K. H.Signatures of links in rational homology spheres. Topology 41 (2002), 1161–1182.Google Scholar
[CK06]Cha, J. C. and Ko, K. H.Signature invariants of covering links. Trans. Amer. Math. Soc. 358 (2006), 3399–3412.Google Scholar
[CK08]Cha, J. C. and Kim, T.Covering link calculus and iterated Bing doubles. Geom. Topol. 12 (2008), 2172–2201.Google Scholar
[CKRS10]Cha, J. C., Kim, T., Ruberman, D. and Strle, S. Smooth concordance of links topologically concordant to the Hopf link. arXiv:1012.2045v1[math.GT] (2010).Google Scholar
[CLR08]Cha, J. C., Livingston, C. and Ruberman, D.Algebraic and Heegaard–Floer invariants of knots with slice Bing doubles. Math. Proc. Camb. Phil. Soc. 144 (2008), 403–410.Google Scholar
[CO90]Cochran, T. D. and Orr, K. E.Not all links are concordant to boundary links. Bulletin Amer. Math. Soc. 23 (1) (1990), 99–106.CrossRefGoogle Scholar
[CO93]Cochran, T. D. and Orr, K. E.Not all links are concordant to boundary links. Ann. Math. 138 (1993), 519–554.Google Scholar
[CO09]Cha, J. C. and Orr, K. E.L(2)-signatures, homology localization and amenable groups. To appear in Comm. Pure Appl. Math. Preprint: arXiv:09103.3700 (2009).Google Scholar
[COT03]Cochran, T. D., Orr, K. E. and Teichner, P.Knot concordance, Whitney towers and L (2) signatures. Ann. of Math. (2) 157 (2003), no. 2, 433–519.Google Scholar
[CST12]Conant, J., Schneiderman, R. and Teichner, P.Whitney tower concordance of classical links. Geom. Topol. 16 (2012), 1419–1479.Google Scholar
[Dav06]Davis, J.A two component link with Alexander polynomial one is concordant to the Hopf link. Math. Proc. Camb. Phil. Soc. 140 (2006), no. 2, 265–268.Google Scholar
[Fox56]Fox, R. H.Free differential calculus. III. Subgroups. Ann. of Math. (2) 64 (1956), 407–419.Google Scholar
[FP12]Friedl, S. and Powell, M.An injectivity theorem for Casson–Gordon type representations relating to the concordance of knots and links. Bull. Korean Math. Soc. 49 (2012), 395–409.Google Scholar
[FQ90]Freedman, M. and Quinn, F.Topology of Four Manifolds (Princeton University Press, 1990).CrossRefGoogle Scholar
[Fre84]Freedman, M. H.The disk theorem for four-dimensional manifolds. Proc. Internat. Congr. Math. vol. 1, 2 (Warsaw, 1983), PWN (1984), pages 647–663.Google Scholar
[Fri03]Friedl, S.Eta invariants as sliceness obstructions and their relation to Casson–Gordon invariants. PhD thesis, Brandeis University (2003).CrossRefGoogle Scholar
[Fri04]Friedl, S.Eta invariants as sliceness obstructions and their relation to Casson–Gordon invariants. Alg. Geom. Topol. 4 (2004), 893–934.CrossRefGoogle Scholar
[Fri05]Friedl, S.Link concordance, boundary link concordance and eta-invariants. Math. Proc. Camb. Phil. Soc. 138 (2005), no. 3, 437–460.CrossRefGoogle Scholar
[Har08]Harvey, S.Homology cobordism invariants and the Cochran–Orr–Teichner filtration of the link concordance group. Geom. Topol. 12 (2008), no. 1, 387–430.Google Scholar
[Hil02]Hillman, J. A.Algebraic Invariants of Links. Series on Knots and Everything 32 (World Scientific Publishing Co. 2002).Google Scholar
[HS97]Hillman, J. A. and Sakuma, M.On the homology of finite abelian coverings of links. Canad. Math. Bull. 40 (1997), 309–315.Google Scholar
[Kaw77]Kawauchi, A.On quadratic forms of 3-manifolds. Invent. Math. 43 (1977), 177–198.CrossRefGoogle Scholar
[Kaw78]Kawauchi, A.On the Alexander polynomials of cobordant links. Osaka J. Math. 15 (1978), no. 1, 151–159.Google Scholar
[Let00]Letsche, C. F.An obstruction to slicing knots using the eta invariant. Math. Proc. Camb. Phil. Soc. 128 (2) (2000), 301–319.Google Scholar
[Lev69]Levine, J.Knot cobordism groups in codimension two. Comment. Math. Helv. 44 (1969), 229–244.Google Scholar
[Lev94]Levine, J.Link invariants via the eta invariant. Comment. Math. Helv. 69 (1994), 82–119.Google Scholar
[Lev07]Levine, J.Concordance of boundary links. J. Knot Theory Ramifications 16 (2007), no. 9, 1111–1120.CrossRefGoogle Scholar
[Lit84]Litherland, R. A. Cobordism of satellite knots. In Four-manifold theory (Durham, N.H., 1982), volume 35 of Contemp. Math., (Amer. Math. Soc. 1984) pages 327–362.Google Scholar
[Mil57]Milnor, J. W. Isotopy of links. Alg. Geom. Top. A symposium in honor of S. Lefschetz, (1957), 280–306.Google Scholar
[MM82]Mayberry, J. P. and Murasugi, K.Torsion-groups of abelian coverings of links. Trans. Amer. Math. Soc. 271 (1982), 143–173.CrossRefGoogle Scholar
[Mur67]Murasugi, K.On a certain numerical invariant of link types. Trans. Amer. Math. Soc. 117 (1967), 387–422.Google Scholar
[Nak78]Nakagawa, Y.On the Alexander polynomials of slice links. Osaka J. Math. 15 (1978), no. 1, 161–182.Google Scholar
[Por04]Porti, J.Mayberry–Murasugi's formula for links in homology 3-spheres. Proc. Amer. Math. Soc. 132 (2004), 3423–3431.Google Scholar
[Sak95]Sakuma, M.Homology of abelian coverings of links and spatial graphs. Canad. J. Math. 47 (1995), 201–224.Google Scholar
[Smo89]Smolinsky, L.Invariants of link cobordism. Proceedings of the 1987 Georgia Topology Conference (Athens, GA, 1987). Topology Appl. 32 (1989), 161–168.Google Scholar
[Sta65]Stallings, J.Homology and central series of groups. J. Algebra 2 (1965), 170–181.CrossRefGoogle Scholar
[Tri69]Tristram, A. G.Some cobordism invariants for links. Proc. Camb. Phil. Soc. 66 (1969), 251–264.Google Scholar
[Tur86]Turaev, V.Reidemeister torsion in knot theory. Russian Math. Surveys 41 (1986), 119–182.CrossRefGoogle Scholar
[Web79]Weber, C.Sur une formule de R. H. Fox concernant l'homologie des revêtements cycliques. Enseign. Math. (2) 25 (1979), 261–272.Google Scholar