Flattening knotted surfaces

Eva Horvat

doi:10.1007/s10711-023-00770-6

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arXiv · 2104.11814

Flattening knotted surfaces

Eva Horvat

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Abstract

A knotted surface in the 4-sphere may be described by means of a hyperbolic diagram that captures the 0-section of a special Morse function, called a hyperbolic decomposition. We show that every hyperbolic decomposition of a knotted surface K defines a projection of K onto a 2-sphere, whose set of critical values is the hyperbolic diagram of K. We apply such projections, called flattenings, to define three invariants of knotted surfaces: the layering, the trunk and the partition number. The basic properties of flattenings and their derived invariants are obtained. Our construction is used to study flattenings of satellite 2-knots.

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Eva Horvat. 2021-04-23. Flattening knotted surfaces. https://doi.org/10.1007/s10711-023-00770-6

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