On knots that divide ribbon knotted surfaces

TL;DR

This paper introduces the concepts of half ribbon genus and half fusion number for knots, computes these invariants for many knots, and explores their relationships with existing knot invariants and properties.

Contribution

It defines new knot invariants related to ribbon 2-knots, computes them for numerous knots, and establishes bounds and differences with classical invariants.

Findings

Computed half ribbon genus for all prime knots up to 12 crossings and many 13-crossing knots.

Introduced the half fusion number and established its lower bound via Levine-Tristram signatures.

Showed that the half fusion number can differ arbitrarily from the standard fusion number.

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.