Flattening knotted surfaces

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.

25 pages, 16 figures

References in corpus (6)

• Bridge trisections of knotted surfaces in $S^4$
• Isotopies of surfaces in 4-manifolds via banded unlink diagrams
• A lower bound on the width of satellite knots
• On a banded link presentation of knotted surfaces
• Widths of surface knots
• Width of codimension two knots