Satellite operators on concordance

Satellite operators on concordance

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Allison Miller, Rice University

Zoom link: https://princeton.zoom.us/j/453512481?pwd=OHZ5TUJvK2trVVlUVmJLZkhIRHFDUT09

The classical satellite construction associates to a pattern knot P in a solid torus and a companion knot K in the 3-sphere a satellite knot P(K), the image of P when the solid torus is ‘tied into’ the knot K. This operation descends to a well-defined map on the set of (smooth or topological) concordance classes of knots. Many natural questions about these maps remain open: when are they surjective, injective, or bijective? How do they behave with respect to measures of 4-dimensional complexity? How do they interact with additional group or metric space structure on the concordance set?

I will discuss work giving partial progress towards answering these questions, including joint work with Piccirillo and with Feller–Pinzon-Caicedo.