Knot concordance, Whitney towers and L 2 -signatures

2003/03/01 by Thomas Cochran, Kent Orr, Kent E. Orr +1 · 18 citations
Mathematics · #Advanced Operator Algebra Research #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.4007/annals.2003.157.433

Abstract

We construct many examples of nonslice knots in 3-space that cannot be distinguished from slice knots by previously known invariants. Using Whitney towers in place of embedded disks, we define a geometric filtration of the 3-dimensional topological knot concordance group. The bottom part of the filtration exhibits all classical concordance invariants, including the Casson-Gordon invariants. As a first step, we construct an infinite sequence of new obstructions that vanish on slice knots. These take values in the L-theory of skew fields associated to certain universal groups. Finally, we use the dimension theory of von Neumann algebras to define an L 2 -signature and use this to detect the first unknown step in our obstruction theory.

Cited by

• A Topological Approach to Cheeger‐Gromov Universal Bounds for von Neumann ρ‐Invariants
• Knot concordance and homology cobordism
• The Non-triviality of the Grope Filtrations of The Knot and Link Concordance Groups
• Unknotted gropes, Whitney towers, and doubly slicing knots
• Cobordisms to weakly splittable links
• Topological minimal genus and L²-signatures
• On the Frøyshov invariant and monopole Lefschetz number
• Reidemeister torsion, the Thurston norm and Harvey's invariants
• A survey of the foundations of four-manifold theory in the topological category
• Structure of the string link concordance group and Hirzebruch-type invariants
• Cut open null-bordisms and derivatives of slice knots
• Polynomial splittings of correction terms and doubly slice knots
• Twisted cohomology pairings of knots I; diagrammatic computation
• Almost-concordance of knots in aspherical 3-manifolds
• Non-commutative Multivariable Reidemeister Torsion and the Thurston Norm
• Blanchfield forms and Gordian distance
• The parity of the Cochran-Harvey invariants of 3-manifolds
• Stabilization and satellite construction of doubly slice links

Related

• Knot concordance, Whitney towers and L² signatures
• Amenable L2-theoretic methods and knot concordance
• A group-valued invariant of knots in the full torus
• Knot concordance and Blanchfield duality
• Filtration of the classical knot concordance group and Casson-Gordon invariants