Gauge Theory and Knot Concordance

Geometry/topology Seminar

Juanita Pinzon Caicedo (North Carolina State University, Mathematics)

Monday, February 19, 2018 -
3:15pm to 4:15pm
Location: 
119 Physics

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 2--dimensional annulus S^1 × [0, 1] into the 4--dimensional cylinder S^3 × [0, 1] that restricts to the given knots at each end. Smooth concordance is an equivalence relation, and the set of smooth concordance classes of knots, C, 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 this talk I will explain how the study of anti-self dual connections on 4--manifolds can be used to shown that (1) the group of topologically slice knots up to smooth concordance contains a subgroup isomorphic to Z^\infty, and (2) satellite operations that are similar to cables are not homomorphisms on C.

Last updated: 2018/11/18 - 2:19pm