A 4-dimensional rational genus bound

TL;DR

This paper introduces a 4-dimensional rational slice genus for knots, establishes lower bounds using Heegaard Floer invariants, and explores implications for Floer simple knots, satellite links, and contact geometry.

Contribution

It defines the rational slice genus, proves bounds via Heegaard Floer invariants, and extends techniques to rational PL slice genus and contact geometry applications.

Findings

Rational slice genus bounds are established using Heegaard Floer $ au$ invariants.

Floer simple knots have rational slice genus equal to their rational Seifert genus.

Sequences of knots with unbounded PL slice genus are constructed.

Abstract

We introduce a 4-dimensional analogue of the rational Seifert genus of a knot $K\subset Y$, which we call the rational slice genus, that measures the complexity of a homology class in $H_2(Y\times [0,1],K;\mathbb{Q})$. Our main theorem is a lower bound for the rational slice genus of a knot in terms of its Heegaard Floer $\tau$ invariants. To prove this, we bound the $\tau$ invariants of any satellite link whose pattern is a closed braid in terms of the $\tau$ invariants of the companion knot, a result which should be of independent value. Our techniques also produce rational PL slice genus bounds. As applications, we use our bounds to prove that Floer simple knots have rational slice genus equal to their rational Seifert genus. We also show that there exist sequences of knots in a fixed 3-manifold whose PL slice genus is unbounded. In addition, we produce stronger bounds for the slice genus of knots relative to the rational longitude, and use these to produce a rational slice-Bennequin bound for knots in contact manifolds with non-trivial contact invariant.