## View abstract

### Session S28 - Knots, Surfaces, 3-manifolds

Friday, July 23, 16:40 ~ 17:10 UTC-3

## Instantons and Knot Concordance

### Juanita Pinzón Caicedo

#### University of Notre Dame, USA   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak0a66f72245b77f6e04fb11ee74c0101c').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy0a66f72245b77f6e04fb11ee74c0101c = 'jp&#105;nz&#111;nc' + '&#64;'; addy0a66f72245b77f6e04fb11ee74c0101c = addy0a66f72245b77f6e04fb11ee74c0101c + 'nd' + '&#46;' + '&#101;d&#117;'; var addy_text0a66f72245b77f6e04fb11ee74c0101c = 'jp&#105;nz&#111;nc' + '&#64;' + 'nd' + '&#46;' + '&#101;d&#117;';document.getElementById('cloak0a66f72245b77f6e04fb11ee74c0101c').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy0a66f72245b77f6e04fb11ee74c0101c + '\'>'+addy_text0a66f72245b77f6e04fb11ee74c0101c+'<\/a>';

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 $\mathcal{C}$ of smooth concordance classes of knots is an abelian group with connected sum as the binary operation. The algebraic structure of $\mathcal{C}$, 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 (1) the group of topologically slice knots up to smooth concordance contains a subgroup isomorphic to $\mathbb{Z}^\infty$, and (2) satellite operations that are similar to cables are not homomorphisms on $\mathcal{C}$.

Joint work with Matt Hedden (Michigan State University, USA) and Tye Lidman (North Carolina State University, USA).