Shadows of 2-knots and complexity

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 $1$.

47 pages, 35 figures

References in corpus (5)

• Bridge trisections of knotted surfaces in $S^4$
• Isotopies of surfaces in 4-manifolds via banded unlink diagrams
• On a banded link presentation of knotted surfaces
• Kirby-Thompson distance for trisections of knotted surfaces
• Shadows of acyclic 4-manifolds with sphere boundary