Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-19T10:47:45.872Z Has data issue: false hasContentIssue false
  • English
  • Français

Homotopy ribbon concordance, Blanchfield pairings, and twisted Alexander polynomials

Part of: Topological manifolds

Published online by Cambridge University Press:  12 April 2021

Stefan Friedl ,
Takahiro Kitayama ,
Lukas Lewark ,
Matthias Nagel  and
Mark Powell
Stefan Friedl
Affiliation:
Department of Mathematics, University of Regensburg, Regensburg, Germany e-mail: sfriedl@gmail.com
Takahiro Kitayama
Affiliation:
Graduate School of Mathematical Sciences, The University of Tokyo, Tokyo, Japan e-mail: kitayama@ms.u-tokyo.ac.jp
Lukas Lewark*
Affiliation:
Department of Mathematics, University of Regensburg, Regensburg, Germany e-mail: sfriedl@gmail.com
Matthias Nagel
Affiliation:
Department of Mathematics, ETH Zurich, Zurich, Switzerland e-mail: matthias.nagel@math.ethz.ch
Mark Powell
Affiliation:
Department of Mathematical Sciences, Durham University, Durham, United Kingdom e-mail: mark.a.powell@durham.ac.uk
*
e-mail: lukas@lewark.de
Get access Rights & Permissions [Opens in a new window]

Abstract

We establish homotopy ribbon concordance obstructions coming from the Blanchfield form and Levine–Tristram signatures. Then, as an application of twisted Alexander polynomials, we show that for every knot K with nontrivial Alexander polynomial, there exists an infinite family of knots that are all concordant to K and have the same Blanchfield form as K, such that no pair of knots in that family is homotopy ribbon concordant.

Keywords

Ribbon concordanceSeifert formBlanchfield pairingtwisted Alexander polynomialLevine–Tristram signatures

MSC classification

Primary: 57N70: Cobordism and concordance
Type
Article
Information
Canadian Journal of Mathematics , Volume 74 , Issue 4 , August 2022 , pp. 1137 - 1176
Check for updates
Copyright
© Canadian Mathematical Society 2021

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

Apostol, T. M., Resultants of cyclotomic polynomials. Proc. Amer. Math. Soc. 24(1970), 457–462.CrossRefGoogle Scholar
Borodzik, M. and Powell, M., Embedded Morse theory and relative splitting of cobordisms of manifolds. J. Geometr. Anal. 26(2016), no. 1, 57–87.CrossRefGoogle Scholar
Bourbaki, N., Éléments de mathématique. Première partie: Les structures fondamentales de l’analyse. Livre II: Algèbre. Chapitre 9: Formes sesquilinéaires et formes quadratiques, Actualités Sci. Ind. no. 1272 Hermann, Paris, 1959, 211 pp.Google Scholar
Cha, J. C. and Friedl, S., Twisted torsion invariants and link concordance. Forum Math. 25(2013), no. 3, 471–504.Google Scholar
Cochran, T. D., Noncommutative knot theory. Algebr. Geom. Topol. 4(2004), 347–398.CrossRefGoogle Scholar
Cochran, T. D., Orr, K. E. and Teichner, P., Knot concordance, Whitney towers and ${L}^2$ -signatures, Ann. of Math. 157(2003), no. 2, 433–519.CrossRefGoogle Scholar
Diederichsen, F. E., Über die Ausreduktion ganzzahliger Gruppendarstellungen bei arithmetischer Äquivalenz (German). Abh. Math. Sem. Hansischen Univ. 13(1940), 357–412.CrossRefGoogle Scholar
Dubois, J., Friedl, S., and Lück, W., Three flavors of twisted invariants of knots. In: Introduction to modern mathematics, Adv. Lect. Math., Vol. 33, Int. Press, Somerville, MA, 2015, pp. 143–169.Google Scholar
Fox, R. H. and Milnor, J. W., Singularities of $2$ -spheres in $4$ -space and cobordism of knots. Osaka J. Math. 3(1966), 257–267.Google Scholar
Freedman, M. H. and Quinn, F., Topology of 4-manifolds. Princeton Mathematical Series, 39, Princeton University Press, Princeton, NJ, 1990.Google Scholar
Friedl, S., Eta invariants as sliceness obstructions and their relation to Casson-Gordon invariants. Algebr. Geom. Topol. 4(2004), 893–934.CrossRefGoogle Scholar
Friedl, S., Leidy, C., Nagel, M., and Powell, M., Twisted Blanchfield pairings and decompositions of 3-manifolds. Homol. Homotopy Appl. 19(2017), no. 2, 275–287.CrossRefGoogle Scholar
Friedl, S., Nagel, M., Orson, P., and Powell, M., A survey of the foundations of four-manifold theory in the topological category. Preprint, 2019. arXiv:1910.07372 Google Scholar
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), no. 2, 395–409.CrossRefGoogle Scholar
Friedl, S. and Powell, M., A calculation of Blanchfield pairings of 3-manifolds and knots. Mosc. Math. J. 17(2017), no. 1, 59–77.CrossRefGoogle Scholar
Friedl, S. and Powell, M., Homotopy ribbon concordance and Alexander polynomials. Archiv der Mathematik, Volume 115 (2020), 717–725.CrossRefGoogle Scholar
Friedl, S. and Vidussi, S., A survey of twisted Alexander polynomials. In: The mathematics of knots, Contrib. Math. Comput. Sci., 1, Springer, Heidelberg, 2011, pp. 45–94.Google Scholar
Gilmer, P. M., Ribbon concordance and a partial order on $S$ -equivalence classes. Topol. Appl. 18(1984), nos. 2–3, 313–324.CrossRefGoogle Scholar
Gordon, C. M. A., Ribbon concordance of knots in the $3$ -sphere. Math. Ann. 257(1981), no. 2, 157–170.CrossRefGoogle Scholar
Herald, C., Kirk, P., and Livingston, C., Metabelian representations, twisted Alexander polynomials, knot slicing, and mutation. Math. Z. 265(2010), no. 4, 925–949.CrossRefGoogle Scholar
Hilton, P. J. and Stammbach, U., A course in homological algebra. Graduate Texts in Mathematics, 4, Springer-Verlag, New York, Berlin, 1971, ix+338 pp.CrossRefGoogle Scholar
Juhász, A., Miller, M., and Zemke, I., Knot cobordisms, bridge index, and torsion in Floer homology. Journal of Topology 13 (2020) 1701–1724.CrossRefGoogle Scholar
Kirk, P. and Livingston, C., Twisted Alexander invariants, Reidemeister torsion, and Casson-Gordon invariants. Topology 38(1999), no. 3, 635–661.CrossRefGoogle Scholar
Kitano, T., Twisted Alexander polynomial and Reidemeister torsion. Pacific J. Math. 174(1996), no. 2, 431–442.CrossRefGoogle Scholar
Lehmer, E. T., A numerical function applied to cyclotomy. Bull. Amer. Math. Soc. 36(1930), no. 4, 291–298.CrossRefGoogle Scholar
Levine, A. S. and Zemke, I., Khovanov homology and ribbon concordances. Bull. Lond. Math. Soc. 51(2019), no. 6, 1099–1103.CrossRefGoogle Scholar
Levine, J., Knot modules. I. Trans. Amer. Math. Soc. 229(1977), 1–50.CrossRefGoogle Scholar
Levine, J., Metabolic and hyperbolic forms from knot theory. J. Pure Appl. Algebra. 58(1989), no. 3, 251–260.CrossRefGoogle Scholar
Lin, X. S., Representations of knot groups and twisted Alexander polynomials. Acta Math. Sin. (Engl. Ser.) 17(2001), no. 3, 361–380.CrossRefGoogle Scholar
Livingston, C., Seifert forms and concordance. Geom. Topol. 6(2002), 403–408.CrossRefGoogle Scholar
Livingston, C. and Moore, A. H., KnotInfo: Table of knot invariants. 2020. www.indiana.edu/~knotinfo Google Scholar
Manolescu, C. and Ozsváth, P., On the Khovanov and knot Floer homologies of quasi-alternating links. Proc. Gökova Geom. Topol. Conf. 2007(2008), 60–81.Google Scholar
Miller, M. and Zemke, I., Knot Floer homology and strongly homotopy-ribbon concordances. Math. Res. Lett. Preprint, 2019. arXiv:1903.05772 Google Scholar
Miyazaki, K., Ribbon concordance does not imply a degree one map. Proc. Amer. Math. Soc. 108(1990), no. 4, 1055–1058.CrossRefGoogle Scholar
Miyazaki, K., Band-sums are ribbon concordant to the connected sum. Proc. Amer. Math. Soc. 126(1998), no. 11, 3401–3406.CrossRefGoogle Scholar
Powell, M., Twisted Blanchfield pairings and symmetric chain complexes. Q. J. Math. 67(2016), no. 4, 715–742.Google Scholar
Ranicki, A., High-dimensional knot theory. Algebraic surgery in codimension 2, with an appendix by E. Winkelnkemper. Springer Monographs in Mathematics, Springer-Verlag, New York, 1998. xxxvi+646 pp.CrossRefGoogle Scholar
Rotman, J. J., An introduction to homological algebra 2nd ed., Universitext, Springer, New York, 2009, xiv+709 pp.Google Scholar
Sarkar, S., Ribbon distance and Khovanov homology. Algebr. Geom. Topol. 20(2020), no. 2, 1041–1058.CrossRefGoogle Scholar
Silver, D. S., On knot-like groups and ribbon concordance. J. Pure Appl. Algebra. 82(1992), no. 1, 99–105.CrossRefGoogle Scholar
Trotter, H. F., On $S$ -equivalence of Seifert matrices. Invent. Math. 20(1973), 173–207.CrossRefGoogle Scholar
Wada, M., Twisted Alexander polynomial for finitely presentable groups. Topology. 33(1994), no. 2, 241–256.CrossRefGoogle Scholar
Zemke, I., Knot Floer homology obstructs ribbon concordance. Ann. Math. 190(2019), no. 3, 931–947.CrossRefGoogle Scholar