A Survey of Classical Knot Concordance

By Charles Livingston. 2003

62 citations recorded by the source corpus.

Publisher Summary This chapter provides an overview of the concordance group of knots in three-dimensional space. It begins with a review of the definitions of knots and concordance and then presents aspects of the algebraic theory of concordance. There are two related theories of concordance, one in the smooth category and the other topological. There are alternative approaches to algebraic obstructions to a knot being slice. A surgery obstruction to finding such a manifold W is determined by the Seifert form, and for a knot of Alexander polynomial one that is the only obstruction, and it vanishes. Although these powerful techniques have revealed a far greater complexity to the concordance group than had been expected, as of now they seem incapable of addressing some of the basic questions: for instance, the slice implies ribbon conjecture and problems related to torsion in the concordance group. Recent work of Cochran, Orr and Teichner has demonstrated a deep structure to the topological concordance group. A discussion of the interplay between three-dimensional knot properties and concordance is also presented in the chapter. The chapter concludes with a brief list of open problems.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/af6c91a24ec1c974f55528796495ecf38557710b

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Citations in the corpus

Knot concordance and higher-order Blanchfield duality (2007, background): “All of these genus one knots are algebraically slice since they have the same Seifert matrix as the slice knot R. Similar knots have appeared in the majority of papers on this subject (for example [ 39 ][38][35][36][19]).”
Sums of lens spaces bounding rational balls (2007, background): “The survey paper of Livingston [8] describes what was known about the group C1 until 2004.”
Cobordism of knots on surfaces (2007, background): “It is natural to compare the knot cobordism defined above with the standard concordance for knots in R3 ; see [12] for a recent survey.”
Metabelian representations, twisted Alexander polynomials, knot slicing, and mutation (2008, background): “See [31] for a list of references up 2003.”
The slice-ribbon conjecture for 3-stranded pretzel knots (2007, unknown): “G T ] 7 A ug 2 00 7”
Knot concordance and von Neumann $\rho$-invariants (2004, methodology): “these were used to show that the kernel of Levine’s map has infinite rank. See [ Li ] for a more detailed history.”
Primary decomposition and the fractal nature of knot concordance (2009, background): “Much work has been done on the subject of knot concordance (for excellent surveys see [26,43]).”