Infection by string links and new structure in the knot concordance group

Burke, John R¹

¹Mathematics and Computer Science Department, Rhode Island College, 600 Mount Pleasant Avenue, Providence, RI 02908-1991, USA

Algebraic and Geometric Topology, Tome 14 (2014) no. 3, pp. 1577-1626

Résumé

This paper highlights the importance of string link concordance in the understanding of knot concordance in general. The results of this paper show that there are infinitely many nontrivial knots in the groups Gn of n–solvable knots modulo n.5–solvable knots, for n greater than or equal to 2, which are not concordant to any knot that is obtained by two or more iterated infections of an Arf invariant zero knot by knots. This latter class accounts for nearly all previously known examples of knots in Gn, n greater than or equal to 2.

In this paper we will generalize the concept of when a rational Laurent polynomial is strongly coprime to another, first introduced by Cochran, Harvey and Leidy, to include multivariable polynomials. We also prove the existence of multivariable polynomials which are strongly coprime to all single variable Laurent polynomials. From this definition of coprimality we define the derived series localized at ℳ for a given sequence of multivariable polynomials ℳ. From such series we obtain refinements of the n–solvable filtration. The operation of infection by a string link is then used to demonstrate that for particular ℳ, certain quotients of successive terms of these refined filtrations have infinite rank.

Détail

DOI : 10.2140/agt.2014.14.1577

Classification : 57M25, 20J05
Keywords: infection, string link, concordance

Bibliographie

[1] A J Casson, C M Gordon, On slice knots in dimension three, from: "Algebraic and geometric topology, Part 2" (editor R J Milgram), Proc. Sympos. Pure Math. 32, Amer. Math. Soc. (1978) 39

[2] A J Casson, C M Gordon, Cobordism of classical knots, from: "Á la recherche de la topologie perdue" (editors L Guillou, A Marin), Progr. Math. 62, Birkhäuser (1986) 181

[3] J C Cha, The structure of the rational concordance group of knots, Mem. Amer. Math. Soc. 885, Amer. Math. Soc. (2007)

[4] J C Cha, Topological minimal genus and $L^2$–signatures, Algebr. Geom. Topol. 8 (2008) 885

[5] T D Cochran, Noncommutative knot theory, Algebr. Geom. Topol. 4 (2004) 347

[6] T D Cochran, S Friedl, P Teichner, New constructions of slice links, Comment. Math. Helv. 84 (2009) 617

[7] T D Cochran, S Harvey, Homology and derived series of groups, II: Dwyer's theorem, Geom. Topol. 12 (2008) 199

[8] T D Cochran, S Harvey, C Leidy, Link concordance and generalized doubling operators, Algebr. Geom. Topol. 8 (2008) 1593

[9] T D Cochran, S Harvey, C Leidy, Knot concordance and higher-order Blanchfield duality, Geom. Topol. 13 (2009) 1419

[10] T D Cochran, S Harvey, C Leidy, Primary decomposition and the fractal nature of knot concordance, Math. Ann. 351 (2011) 443

[11] T D Cochran, K E Orr, P Teichner, Knot concordance, Whitney towers and $L^2$–signatures, Ann. of Math. 157 (2003) 433

[12] T D Cochran, K E Orr, P Teichner, Structure in the classical knot concordance group, Comment. Math. Helv. 79 (2004) 105

[13] T D Cochran, P Teichner, Knot concordance and von Neumann $\rho$–invariants, Duke Math. J. 137 (2007) 337

[14] R H Fox, J W Milnor, Singularities of $2$–spheres in $4$–space and cobordism of knots, Osaka J. Math. 3 (1966) 257

[15] J A Hillman, Alexander ideals of links, Lecture Notes in Mathematics 895, Springer (1981)

[16] B J Jiang, A simple proof that the concordance group of algebraically slice knots is infinitely generated, Proc. Amer. Math. Soc. 83 (1981) 189

[17] M A Kervaire, J W Milnor, On $2$–spheres in $4$–manifolds, Proc. Nat. Acad. Sci. U.S.A. 47 (1961) 1651

[18] S G Kim, Polynomial splittings of Casson–Gordon invariants, Math. Proc. Cambridge Philos. Soc. 138 (2005) 59

[19] S G Kim, T Kim, Polynomial splittings of metabelian von Neumann rho-invariants of knots, Proc. Amer. Math. Soc. 136 (2008) 4079

[20] C Leidy, Higher-order linking forms, PhD thesis, Rice University (2004)

[21] C Leidy, Higher-order linking forms for knots, Comment. Math. Helv. 81 (2006) 755

[22] J Levine, Knot cobordism groups in codimension two, Comment. Math. Helv. 44 (1969) 229

[23] J Levine, Knot modules, I, Trans. Amer. Math. Soc. 229 (1977) 1

[24] J Levine, Signature invariants of homology bordism with applications to links, from: "Knots 90" (editor A Kawauchi), de Gruyter (1992) 395

[25] D S Passman, The algebraic structure of group rings, Robert E Krieger Publishing Co. (1985)

[26] D Rolfsen, Knots and links, Mathematics Lecture Series 7, Publish or Perish (1990)

[27] N W Stoltzfus, Unraveling the integral knot concordance group, Mem. Amer. Math. Soc. 192, Amer. Math. Soc. (1977)