On a banded link presentation of knotted surfaces

TL;DR

This paper introduces a visual presentation method for knotted surfaces in four-dimensional space using Morse critical points, and explores surface-knots with specific critical point configurations, including a diagrammatic representation of twist-spun torus knots.

Contribution

It presents a novel banded link presentation method for knotted surfaces, analyzes surface-knots with particular critical points, and provides diagrammatic forms for complex surface-knots.

Findings

Existence of infinitely many distinct surface-knots with two index 1 critical points

A diagrammatic long flat form for n-twist-spun (2,t)-torus knots

A method for visualizing knotted surfaces via Morse critical points

Abstract

We will discuss a method for visual presentation of knotted surfaces in the four space, by examining a number and a position of its Morse's critical points. Using this method, we will investigate surface-knot with one critical point of index 1. Then we show infinitely many mutually distinct surface-knots that has an embedding with two critical points of index 1. Next we define a long flat for a banded link for any surface-knot and show diagrammatically a long flat form of $n$-twist-spun $(2,t)$-torus knots.