Knots are smooth 1-dimensional submanifolds of the 3-dimensional sphere S^3, diffeomorphic to the circle S^1, which are usually studied up to ambient isotopy. A natural generalization in dimension 4 of the question whether certain knots are isotopic to the trivial knot is the concept of concordance, an equivalence relation on the set of all knots.
In my PhD thesis, I showed that every non-trivial strongly quasipositive knot is concordant to infinitely many pairwise non-isotopic strongly quasipositive knots. In contrast, Baker conjectured that strongly quasipositive fibered knots are isotopic if they are concordant. Our construction uses a satellite operation with companion a slice knot with maximal Thurston–Bennequin number –1. In joint work with Maciej Borodzik and Hannah Turner, using this construction and Khovanov homology as an obstruction to ribbon concordance, we found a pair of strongly quasipositive knots that are concordant but not ribbon concordant in either direction.
In the talk, we will define the relevant terms necessary to understand these results and explain their context.
