Algebraic & Geometric Topology

Derivatives of knots and second-order signatures

Tim D Cochran, Shelly Harvey, and Constance Leidy

Full-text: Open access


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.

Article information

Algebr. Geom. Topol., Volume 10, Number 2 (2010), 739-787.

Received: 29 December 2008
Revised: 17 January 2010
Accepted: 17 January 2010
First available in Project Euclid: 19 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}
Secondary: 57M10: Covering spaces

knot concordance slice knot $n$–solvable signature


Cochran, Tim D; Harvey, Shelly; Leidy, Constance. Derivatives of knots and second-order signatures. Algebr. Geom. Topol. 10 (2010), no. 2, 739--787. doi:10.2140/agt.2010.10.739.

Export citation


  • 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”, (R J Milgram, editor), Proc. Sympos. Pure Math. XXXII, Amer. Math. Soc. (1978) 39–53
  • A J Casson, C M Gordon, Cobordism of classical knots, from: “À la recherche de la topologie perdue”, (L Guillou, A Marin, editors), Progr. Math. 62, Birkhäuser, Boston (1986) 181–199 With an appendix by P M Gilmer
  • J Cheeger, M Gromov, Bounds on the von Neumann dimension of $L\sp 2$–cohomology and the Gauss–Bonnet theorem for open manifolds, J. Differential Geom. 21 (1985) 1–34
  • D Cimasoni, V Florens, Generalized Seifert surfaces and signatures of colored links, Trans. Amer. Math. Soc. 360 (2008) 1223–1264
  • T D Cochran, Noncommutative knot theory, Algebr. Geom. Topol. 4 (2004) 347–398
  • T D Cochran, S Friedl, P Teichner, New constructions of slice links, Comment. Math. Helv. 84 (2009) 617–638
  • T D Cochran, S Harvey, Homology and derived $p$–series of groups, J. Lond. Math. Soc. $(2)$ 78 (2008) 677–692
  • T D Cochran, S Harvey, C Leidy, Knot concordance and Blanchfield duality
  • T D Cochran, S Harvey, C Leidy, Knot concordance and Blanchfield duality, Oberwolfach Reports 3 (2006)
  • T D Cochran, S Harvey, C Leidy, Link concordance and generalized doubling operators, Algebr. Geom. Topol. 8 (2008) 1593–1646
  • T D Cochran, S Harvey, C Leidy, Knot concordance and higher-order Blanchfield duality, Geom. Topol. 13 (2009) 1419–1482
  • T D Cochran, K E Orr, Homology boundary links and Blanchfield forms: concordance classification and new tangle-theoretic constructions, Topology 33 (1994) 397–427
  • T D Cochran, K E Orr, P Teichner, Knot concordance, Whitney towers and $L\sp 2$–signatures, Ann. of Math. $(2)$ 157 (2003) 433–519
  • T D Cochran, K E Orr, P Teichner, Structure in the classical knot concordance group, Comment. Math. Helv. 79 (2004) 105–123
  • T D Cochran, P Teichner, Knot concordance and von Neumann $\rho$–invariants, Duke Math. J. 137 (2007) 337–379
  • S Friedl, Eta invariants as sliceness obstructions and their relation to Casson-Gordon invariants, Algebr. Geom. Topol. 4 (2004) 893–934
  • P Gilmer, Slice knots in $S\sp{3}$, Quart. J. Math. Oxford Ser. $(2)$ 34 (1983) 305–322
  • 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) 895–945
  • J A Hillman, Alexander ideals of links, Lecture Notes in Math. 895, Springer, Berlin (1981)
  • P D Horn, Higher-order genera of knots
  • A Kawauchi, A survey of knot theory, Birkhäuser Verlag, Basel (1996) Translated and revised from the 1990 Japanese original by the author
  • C Kearton, Cobordism of knots and Blanchfield duality, J. London Math. Soc. $(2)$ 10 (1975) 406–408
  • T Kim, An infinite family of non-concordant knots having the same Seifert form, Comment. Math. Helv. 80 (2005) 147–155
  • C Lamm, Symmetric unions and ribbon knots, Osaka J. Math. 37 (2000) 537–550
  • C Leidy, Higher-order linking forms for $3$–manifolds, preprint
  • C F Letsche, An obstruction to slicing knots using the eta invariant, Math. Proc. Cambridge Philos. Soc. 128 (2000) 301–319
  • J Levine, Knot cobordism groups in codimension two, Comment. Math. Helv. 44 (1969) 229–244
  • W Lück, T Schick, Various $L\sp 2$–signatures and a topological $L\sp 2$–signature theorem, from: “High-dimensional manifold topology”, (F T Farrell, W Lück, editors), World Sci. Publ., River Edge, NJ (2003) 362–399
  • L Smolinsky, Invariants of link cobordism, from: “Proceedings of the 1987 Georgia Topology Conference (Athens, GA, 1987)”, (N Habegger, C McCrory, editors), volume 32 (1989) 161–168
  • R Strebel, Homological methods applied to the derived series of groups, Comment. Math. Helv. 49 (1974) 302–332