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Whitehead Doubles and Non-Orientable Surfaces

Megan Fairchild

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.

Whitehead Doubles and Non-Orientable Surfaces

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 -twisted Whitehead doubles and provide genus 1 non-orientable cobordisms to cable knots.

Paper Structure

This paper contains 9 sections, 26 theorems, 22 equations, 8 figures.

Table of Contents

  1. INTRODUCTION
  2. THE SLICE PROBLEM
  3. Whitehead Double
  4. Classical Knot Invariants
  5. Knot Floer Homology
  6. A 4-Manifold Obstruction
  7. NON-ORIENTABLE COBORDISMS
  8. Cables and more Knot Floer Homology
  9. NON-ORIENTABLE SLICENESS

Key Result

Theorem 1.1

The $t$-twisted Whitehead double operation does not preserve non-orientable sliceness.

Figures (8)

  • Figure 2.1: The Whitehead Double Pattern
  • Figure 2.2:
  • Figure 3.1: Non-Orientable Band Move
  • Figure 3.2: Non-Orientable Band Move from the figure 8 knot to the trefoil
  • Figure 3.3: Non-Orientable Band Move
  • ...and 3 more figures

Theorems & Definitions (39)

  • Theorem 1.1
  • Conjecture 1.2: Problem 1.38 in Kirby1995ProblemsIL
  • Theorem 1.3
  • Remark 2.1
  • Theorem 2.2: Theorem 8.3 in signature
  • Example 2.3
  • Theorem 2.4: Fox-Milnor Condition FoxMilnor
  • Remark 2.5
  • Remark 2.6
  • Remark 2.7
  • ...and 29 more