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Unknotting Nonorientable Surfaces of Genus 4 and 5

Mark Pencovitch

Abstract

Expanding on work by Conway, Orson, and Powell, we study the isotopy classes rel. boundary of nonorientable, compact, locally flatly embedded surfaces in $D^4$ with knot group $\mathbb{Z}_2$. In particular we show that if two such surfaces have fixed knot boundary $K$ in $S^4$ such that $\vert \det(K) \vert =1$, the same normal Euler number, and the same nonorientable genus $4$ or $5$, then they are ambiently isotopic rel. boundary. This implies that closed, nonorientable, locally flatly embedded surfaces in the $4$-sphere with knot group $\mathbb{Z}_2$ of nonorientable genus $4$ and $5$ are topologically unknotted. The proof relies on calculations, implemented in Sage, which imply that the modified surgery obstruction is elementary. Furthermore we show that this method fails for nonorientable genus $6$ and $7$.

Unknotting Nonorientable Surfaces of Genus 4 and 5

Abstract

Expanding on work by Conway, Orson, and Powell, we study the isotopy classes rel. boundary of nonorientable, compact, locally flatly embedded surfaces in with knot group . In particular we show that if two such surfaces have fixed knot boundary in such that , the same normal Euler number, and the same nonorientable genus or , then they are ambiently isotopic rel. boundary. This implies that closed, nonorientable, locally flatly embedded surfaces in the -sphere with knot group of nonorientable genus and are topologically unknotted. The proof relies on calculations, implemented in Sage, which imply that the modified surgery obstruction is elementary. Furthermore we show that this method fails for nonorientable genus and .
Paper Structure (12 sections, 12 theorems, 26 equations)

This paper contains 12 sections, 12 theorems, 26 equations.

Table of Contents

  1. Introduction
  2. Background
  3. The modified surgery obstruction
  4. Quadratic forms
  5. Calculating triviality
  6. Explicit Calculations
  7. Calculation for $h=2$
  8. Calculation for $h=3$
  9. Calculation for $h=4$
  10. Calculation for $h=5$
  11. Comments and Conclusions
  12. Description of the Sage Code

Key Result

Theorem 1.1

Let $F_0,F_1$ be $\mathbb{Z}_2$-surfaces in $D^4$ of nonorientable genus $h$, boundary $K$, and the same extremal normal Euler number. Assume that $\vert \det(K) \vert =1$. If $h = 4,5$ then $F_0$ and $F_1$ are ambiently isotopic rel. boundary. In particular, every $\mathbb{Z}_2$-surface in $S^4$ wi

Theorems & Definitions (33)

  • Definition 1.1
  • Theorem 1.1
  • Conjecture 1.2
  • Definition 2.1
  • Theorem 2.1: Kreck Kreck
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • ...and 23 more