Luka Marčič: Ribbon concordance quandles, part 2

The notions of slice knots and concordance between knots lies in the intersection of knot theory and the topology of 3 and 4 manifolds. Introduced by Fox and Milnor in the 60s, it has since become a very active area of research with many deep-rooted still unanswered questions. Until recently, one such question was whether ribbon concordance induces a partial order on the set of knots and on the knot concordance group. Posed by Gordon in his 1981 article where he introduced the notion of ribbon concordance, it was only answered some 40 years later in 2022 by Agol after a resurgence of study of effects of ribbon concordances on knot invariants. On the other hand, in 1982 David Joyce defined quandles, algebraic structures modeled after knots, and showed that they are an almost complete knot invariant, albeit an unpractical one. Fenn and Rourke then defined quandles for an arbitrary codimension 2 embedding between manifolds in 1992. Since the turn of the century, quandle research became more prominent, greatly expanding our algebraic understanding of the structure and giving us a myriad of new knot and surface-knot invariants. In the first talk, we start with an overview of classical theory of knot concordance: slice knots, concordance group, ribbon knots, ribbon concordance and the slice-ribbon conjecture. We then continue and end with quandles, specifically with representations of fundamental quandles of knots and of knotted surfaces. In the second talk, we showcase a theorem proved by Eva Horvat and the speaker which connects the fundamental quandle of two ribbon concordant knots with the fundamental quandle of the ribbon concordance itself and is analogous to a similar statement about fundamental groups proved by Gordon in his original article.