On knots that divide ribbon knotted surfaces

TL;DR

This work investigates knots that divide ribbon surface-knots by introducing the half ribbon genus $g_{hr}(K)$ and the half fusion number $f_h(K)$, situating them between slice and double-slice notions. It develops a framework tying band attachments to 1-handle additions on dividing surface-knots to derive genus bounds that relate $g_4$, $g_{hr}$, and $g_{ds}$. The authors compute $g_{hr}$ for all prime knots with $ ext{crossings} \,\le\ 12$ (finding $g_{hr}(K)=2g_4(K)$ in these cases) and obtain several new double-slice genera, while also exhibiting arbitrarily large gaps between half ribbon and double slice genera via explicit constructions. They also define the half fusion number $f_h(K)$, bound it by Levine-Tristram signatures, and show that $f_h(K)$ can differ arbitrarily from the standard fusion number, accompanied by extensive computations and data releases for the community.

Abstract

We define a knot to be half ribbon if it is the cross-section of a ribbon 2-knot, and observe that ribbon implies half ribbon implies slice. We introduce the half ribbon genus of a knot K, the minimum genus of a ribbon knotted surface of which K is a cross-section. We compute this genus for all prime knots up to 12 crossings, and many 13-crossing knots. The same approach yields new computations of the doubly slice genus. We also introduce the half fusion number of a knot K, that measures the complexity of ribbon 2-knots of which K is a cross-section. We show that it is bounded from below by the Levine-Tristram signatures, and differs from the standard fusion number by an arbitrarily large amount.

Figures (5)

• Figure 1: On the left: a ribbon surface $F$ formed of discs and bands. On the right: the induced sphere-tube presentation of the double of $F$.
• Figure 2: Schematic diagrams of the split union of $S_1$ and $S_2$, and the surface-knots $S'$ and $S^{\prime\prime}$.
• Figure 3: Attaching $1$-handles with cores given by the dashed arcs introduces an unlink to the equatorial cross-section.
• Figure 4: On the left: bands defining a ribbon cobordism $C$. On the right: a schematic of $C \cup \overline{C}$ (some handles have been isotoped away from the equator for aesthetic purposes.)
• Figure 5: A diagram of the $1$-knot $10_{74}$, together with three bands. Attaching the bands labelled $A$ and $B$ realises a crossing change that yields a diagram of the $1$-knot $9_{46}$. Subsequently attaching the band labelled $C$ defines a ribbon disc for $9_{46}$.

Theorems & Definitions (20)

• Theorem 1
• Proposition 2
• proof
• Proposition 3
• proof
• Proposition 4
• proof
• Theorem 5
• proof
• Proposition 6