Shadows of 2-knots and complexity

TL;DR

This paper introduces the shadow-complexity invariant for 2-knots in 4-spheres, characterizes those with minimal complexity, and demonstrates the existence of infinitely many knots with shadow-complexity one.

Contribution

It defines a new invariant for 2-knots based on Turaev shadows and characterizes knots with shadow-complexity at most one.

Findings

Unknot is the only 2-knot with shadow-complexity 0.

Infinitely many 2-knots have shadow-complexity 1.

Abstract

We introduce a new invariant for a $2$-knot in $S^4$, called the shadow-complexity, based on the theory of Turaev shadows, and we give a characterization of $2$-knots with shadow-complexity at most $1$. Specifically, we show that the unknot is the only $2$-knot with shadow-complexity $0$ and that there exist infinitely many $2$-knots with shadow-complexity $1$.