Whitehead Doubles and Non-Orientable Surfaces

TL;DR

This paper explores the properties of Whitehead doubles, especially the Figure 8 knot, focusing on their sliceness and non-orientable genus, and introduces new bounds and cobordism constructions.

Contribution

It provides new bounds for the non-orientable 4 genus of t-twisted Whitehead doubles and constructs genus 1 non-orientable cobordisms to cable knots.

Findings

Whitehead doubles can be topologically slice but not smoothly slice.

Improved bounds for the non-orientable 4 genus of t-twisted Whitehead doubles.

Constructed genus 1 non-orientable cobordisms to cable knots.

Abstract

Whitehead doubles provide a plethora of examples of knots that are topologically slice but not smoothly slice. We discuss the problem of the Whitehead double of the Figure 8 knot and survey commonly used techniques to obstructing sliceness. Additionally, we improve bounds in general for the non-orientable 4 genus of $t$-twisted Whitehead doubles and provide genus 1 non-orientable cobordisms to cable knots.