Instantons and knot concordance

Abstract: Knot concordance can be regarded as the study of knots as boundaries of surfaces embedded in spaces of dimension 4. Specifically, two knots $K_0$ and $K_1$ are said to be smoothly concordant if there is a smooth embedding of the annulus $S^1 \times [0, 1]$ into the “cylinder” $S^3 \times [0, 1]$ that restricts to the given knots at each end. Smooth concordance is an equivalence relation, and the set C of smooth concordance classes of knots is an abelian group with connected sum as the binary operation. The algebraic structure of $C$, the concordance class of the unknot, and the set of knots that are topologically slice but not smoothly slice are much studied objects in low-dimensional topology. Gauge theoretical results on the nonexistence of certain definite smooth 4-manifolds can be used to better understand these objects. In particular, the study of anti-self dual connections on 4-manifolds can be used to shown that the group of topologically slice knots up to smooth concordance contains a subgroup isomorphic to $Z^\infty.$

general topologygeometric topology

Audience: researchers in the discipline

K-State Mathematics Department Women Lecture Series