Quadratic forms in knot theory (Q2704202)

This is a well-written survey article explaining how the theory of quadratic and Hermitian forms allows to obtain geometric results in the theory of knots of codimension two. The paper starts by introducing the notions of Seifert surface, Seifert matrix, \(S\)-equivalence, Blanchfield pairing, knot cobordism. The set of isotopy classes of \(n\)-dimensional knots of codimension two \((S^{n+2},k^n)\) is an abelian semigroup with respect to the operation of connected sum of knots. The map, which assigns to a knot its Blanchfield form, is a homomorphism of this semigroup into a semigroup of Hermitian forms. This correspondence is an isomorphism for simple odd-dimensional knots (J. Levine, C. Kearton). The results about non-unique factorization into irreducible knots and failure of the cancellation property for high-dimensional knots \((n>2)\) are presented. These results were originally obtained by E. Bayer-Fluckiger, J. Hillman, C. Kearton and S. M. J. Willson. For the classical knots (i.e. \(n=1)\) H. Schubert proved that the factorization into simple irreducible knots is unique.NEWLINENEWLINEFor the entire collection see [Zbl 0956.00036].