Mathematics Colloquia and Seminars

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Two knots $K_0$ and $K_1$ are said to be smoothly concordant if the connected sum $K_0\#m(K_1^r)$ bounds a disk smoothly embedded in the 4-ball. 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. Satellite operations, or the process of tying a given knot P along another knot K to produce a third knot P(K), are powerful tools for studying the algebraic structure of the concordance group. In this talk I will describe conditions on the pattern P that suffice to conclude that the function $P:\mathcal{C}\to \mathcal{C}$ is not a homomorphism. This is joint work with Tye Lidman and Allison Miller.