Widths of surface knots

TL;DR

This paper investigates the numerical invariant called width for surface knots in 4-space, using planar projections with singularities, and explores its relation to the surface braid index, providing classifications for knots with small widths.

Contribution

It introduces the concept of width for surface knots, computes widths for specific examples, and relates this invariant to the surface braid index.

Findings

Determined widths of certain surface knots.

Characterized surface knots with small total widths.

Explored the relation between width and surface braid index.

Abstract

We study surface knots in 4-space by using generic planar projections. These projections have fold points and cusps as their singularities and the image of the singular point set divides the plane into several regions. The width (or the total width) of a surface knot is a numerical invariant related to the number of points in the inverse image of a point in each of the regions. We determine the widths of certain surface knots and characterize those surface knots with small total widths. Relation to the surface braid index is also studied.