Derivatives of knots and second-order signatures

Cochran, Tim D¹ ; Harvey, Shelly¹ ; Leidy, Constance²

¹Department of Mathematics MS-136, Rice University, PO 1892, Houston, Texas 77251-1892, USA
²Department of Mathematics, Wesleyan University, Wesleyan Station, Middletown, CT 06459, USA

Algebraic and Geometric Topology, Tome 10 (2010) no. 2, pp. 739-787

Résumé

We define a set of “second-order” L(2)–signature invariants for any algebraically slice knot. These obstruct a knot’s being a slice knot and generalize Casson–Gordon invariants, which we consider to be “first-order signatures”. As one application we prove: If K is a genus one slice knot then, on any genus one Seifert surface Σ, there exists a homologically essential simple closed curve J of self-linking zero, which has vanishing zero-th order signature and a vanishing first-order signature. This extends theorems of Cooper and Gilmer. We introduce a geometric notion, that of a derivative of a knot with respect to a metabolizer. We also introduce a new relation, generalizing homology cobordism, called null-bordism.

Détail

DOI : 10.2140/agt.2010.10.739

Keywords: knot concordance, slice knot, $n$–solvable, signature

Bibliographie

[1] A J Casson, C M Gordon, On slice knots in dimension three, from: "Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., 1976), Part 2" (editor R J Milgram), Proc. Sympos. Pure Math. XXXII, Amer. Math. Soc. (1978) 39

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

[3] J Cheeger, M Gromov, Bounds on the von Neumann dimension of $L^2$–cohomology and the Gauss–Bonnet theorem for open manifolds, J. Differential Geom. 21 (1985) 1

[4] D Cimasoni, V Florens, Generalized Seifert surfaces and signatures of colored links, Trans. Amer. Math. Soc. 360 (2008) 1223

[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 $p$–series of groups, J. Lond. Math. Soc. $(2)$ 78 (2008) 677

[8] T D Cochran, S Harvey, C Leidy, Knot concordance and Blanchfield duality

[9] T D Cochran, S Harvey, C Leidy, Knot concordance and Blanchfield duality, Oberwolfach Reports 3 (2006)

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

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

[12] T D Cochran, K E Orr, Homology boundary links and Blanchfield forms: concordance classification and new tangle-theoretic constructions, Topology 33 (1994) 397

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

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

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

[16] S Friedl, Eta invariants as sliceness obstructions and their relation to Casson-Gordon invariants, Algebr. Geom. Topol. 4 (2004) 893

[17] P Gilmer, Slice knots in $S^{3}$, Quart. J. Math. Oxford Ser. $(2)$ 34 (1983) 305

[18] P Gilmer, Classical knot and link concordance, Comment. Math. Helv. 68 (1993) 1

[19] S Harvey, Higher-order polynomial invariants of $3$–manifolds giving lower bounds for the Thurston norm, Topology 44 (2005) 895

[20] J A Hillman, Alexander ideals of links, Lecture Notes in Math. 895, Springer (1981)

[21] P D Horn, Higher-order genera of knots

[22] A Kawauchi, A survey of knot theory, Birkhäuser Verlag (1996)

[23] C Kearton, Cobordism of knots and Blanchfield duality, J. London Math. Soc. $(2)$ 10 (1975) 406

[24] T Kim, An infinite family of non-concordant knots having the same Seifert form, Comment. Math. Helv. 80 (2005) 147

[25] C Lamm, Symmetric unions and ribbon knots, Osaka J. Math. 37 (2000) 537

[26] C Leidy, Higher-order linking forms for $3$–manifolds, preprint

[27] C F Letsche, An obstruction to slicing knots using the eta invariant, Math. Proc. Cambridge Philos. Soc. 128 (2000) 301

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

[29] W Lück, T Schick, Various $L^2$–signatures and a topological $L^2$–signature theorem, from: "High-dimensional manifold topology" (editors F T Farrell, W Lück), World Sci. Publ. (2003) 362

[30] L Smolinsky, Invariants of link cobordism, from: "Proceedings of the 1987 Georgia Topology Conference (Athens, GA, 1987)" (editors N Habegger, C McCrory) (1989) 161

[31] R Strebel, Homological methods applied to the derived series of groups, Comment. Math. Helv. 49 (1974) 302