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Topology/Geometry Seminar

Satellite knots and immersed curves

Jonathan Hanselman (Princeton University)

Location:  Hill 705
Date & time: Tuesday, 09 April 2024 at 4:00PM - 5:00PM

Satellite operations are a valuable method of constructing complicated knots from simpler ones, and much work has gone into understanding how various knot invariants change under these operations. We describe a new way of computing the (UV=0 quotient of the) knot Floer complex using an immersed Heegaard diagram obtained from a Heegaard diagram for the pattern and the immersed curve representing the UV=0 knot Floer complex of the companion. This is particularly useful for (1,1)-patterns, since in this case the resulting immersed diagram is genus one and the computation is combinatorial. In the case of one-bridge braid satellites the immersed curve invariant for the satellite can be obtained directly from that of the companion by deforming the diagram, generalizing earlier work with Watson on cables. This is joint work with Wenzhao Chen.

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