Algebraic & Geometric Topology

The Milnor triple linking number of string links by cut-and-paste topology

Robin Koytcheff

Full-text: Open access


Bott and Taubes constructed knot invariants by integrating differential forms along the fiber of a bundle over the space of knots, generalizing the Gauss linking integral. Their techniques were later used to construct real cohomology classes in spaces of knots and links in higher-dimensional Euclidean spaces. In previous work, we constructed cohomology classes in knot spaces with arbitrary coefficients by integrating via a Pontrjagin–Thom construction. We carry out a similar construction over the space of string links, but with a refinement in which configuration spaces are glued together according to the combinatorics of weight systems. This gluing is somewhat similar to work of Kuperberg and Thurston. We use a formula of Mellor for weight systems of Milnor invariants, and we thus recover the Milnor triple linking number for string links, which is in some sense the simplest interesting example of a class obtained by this gluing refinement of our previous methods. Along the way, we find a description of this triple linking number as a degree of a map from the 6–sphere to a quotient of the product of three 2–spheres.

Article information

Algebr. Geom. Topol., Volume 14, Number 2 (2014), 1205-1247.

Received: 7 March 2013
Revised: 30 August 2013
Accepted: 20 October 2013
First available in Project Euclid: 19 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55R80: Discriminantal varieties, configuration spaces 57M27: Invariants of knots and 3-manifolds
Secondary: 55R12: Transfer 57R40: Embeddings

triple linking number configuration space integrals Pontrjagin–Thom construction gluing degree string links finite-type link invariants


Koytcheff, Robin. The Milnor triple linking number of string links by cut-and-paste topology. Algebr. Geom. Topol. 14 (2014), no. 2, 1205--1247. doi:10.2140/agt.2014.14.1205.

Export citation


  • J W Auer, Fiber integration and Poincaré duality, Tensor 33 (1979) 72–74
  • S Axelrod, I M Singer, Chern–Simons perturbation theory, II, J. Differential Geom. 39 (1994) 173–213
  • D Bar-Natan, On the Vassiliev knot invariants, Topology 34 (1995) 423–472
  • D Bar-Natan, Vassiliev homotopy string link invariants, J. Knot Theory Ramifications 4 (1995) 13–32
  • R Bott, C Taubes, On the self-linking of knots: Topology and physics, J. Math. Phys. 35 (1994) 5247–5287
  • R Bott, L W Tu, Differential forms in algebraic topology, Graduate Texts in Mathematics 82, Springer, New York (1982)
  • A S Cattaneo, P Cotta-Ramusino, R Longoni, Configuration spaces and Vassiliev classes in any dimension, Algebr. Geom. Topol. 2 (2002) 949–1000
  • D DeTurck, H Gluck, R Komendarczyk, P Melvin, C Shonkwiler, D S Vela-Vick, Triple linking numbers, ambiguous Hopf invariants and integral formulas for three-component links, Mat. Contemp. 34 (2008) 251–283
  • W Fulton, R MacPherson, A compactification of configuration spaces, Ann. of Math. 139 (1994) 183–225
  • W Greub, S Halperin, R Vanstone, Connections, curvature, and cohomology, Vol. I: De Rham cohomology of manifolds and vector bundles, Pure and Applied Mathematics 47, Academic Press, New York (1972)
  • N Habegger, X-S Lin, The classification of links up to link-homotopy, J. Amer. Math. Soc. 3 (1990) 389–419
  • K Jänich, On the classification of $O(n)$–manifolds, Math. Ann. 176 (1968) 53–76
  • D Joyce, On manifolds with corners, from: “Advances in geometric analysis”, (S Janeczko, J Li, D H Phong, editors), Advanced Lectures in Mathematics 21, International Press, Boston, MA (2012) 225–258
  • M Kontsevich, Feynman diagrams and low-dimensional topology, from: “First European Congress of Mathematics, Vol. II”, (A Joseph, F Mignot, F Murat, B Prum, R Rentschler, editors), Progr. Math. 120, Birkhäuser, Basel (1994) 97–121
  • R Koytcheff, A homotopy-theoretic view of Bott–Taubes integrals and knot spaces, Algebr. Geom. Topol. 9 (2009) 1467–1501
  • R Koytcheff, B Munson, I Volić, Configuration space integrals and the cohomology of the space of homotopy string links (2013)
  • G Kuperberg, D Thurston, Pertubative $3$–manifold invariants by cut-and-paste topology (2000)
  • G Laures, On cobordism of manifolds with corners, Trans. Amer. Math. Soc. 352 (2000) 5667–5688
  • C Lescop, On the Kontsevich–Kuperberg–Thurston construction of a configuration-space invariant for rational homology $3$–spheres, Prépublication $655$, Institut Fourier (2004) Available at \setbox0\makeatletter\@url {\unhbox0
  • B Mellor, Weight systems for Milnor invariants, J. Knot Theory Ramifications 17 (2008) 213–230
  • B Mellor, P Melvin, A geometric interpretation of Milnor's triple linking numbers, Algebr. Geom. Topol. 3 (2003) 557–568
  • J Milnor, Link groups, Ann. of Math. 59 (1954) 177–195
  • J Milnor, Isotopy of links: Algebraic geometry and topology, from: “A symposium in honor of S Lefschetz”, (R H Fox, D C Spencer, A W Tucker, editors), Princeton Univ. Press (1957) 280–306
  • K Pelatt, A geometric homology representative in the space of long knots
  • M Polyak, Skein relations for Milnor's $\mu$–invariants, Algebr. Geom. Topol. 5 (2005) 1471–1479
  • V V Prasolov, A B Sossinsky, Knots, links, braids and $3$–manifolds: An introduction to the new invariants in low-dimensional topology, Translations of Mathematical Monographs 154, Amer. Math. Soc. (1997)
  • D Thurston, Integral expressions for the Vassiliev knot invariants, AB thesis, Harvard University (1995)
  • V Turchin, Context-free manifold calculus and the Fulton–MacPherson operad, Algebr. Geom. Topol. 13 (2013) 1243–1271
  • I Volić, A survey of Bott–Taubes integration, J. Knot Theory Ramifications 16 (2007) 1–42