# Seifert solids and tri-plane diagrams

### Abstract

Whereas every knot in $S^3$ bounds a Seifert surface, every orientable knotted surface in $S^4$ bounds an embedded 3-manifold called a Seifert solid. Recently, we showed that every knotted surface can be represented by a tri-plane diagram, a triple of trivial tangle diagrams that join in pairs to form unlink diagrams. We describe a procedure that uses a tri-plane diagram for a knotted surface $K$ to produce a Seifert solid for $K$. Notably, this procedure restricts to the classical Seifert’s algorithm on each paired unlink diagram. This is joint work with Jason Joseph, Jeffrey Meier, and Maggie Miller.

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