Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Extending Quotients of Knot Groups over Surfaces in $B^4$

Alexandra Kjuchukova, Kent E. Orr

Abstract

Let $K\subseteq S^3$ be a knot with exterior $E_K$, and denote by $ρ\colon π_1(E_K)\twoheadrightarrow G$ a quotient of its group. We give a sharp obstruction to the existence of a connected, oriented, smooth surface $F\subseteq B^4$ with $\partial F = K$ over whose exterior $ρ$ extends surjectively. Equivalently, we determine whether the cover of $S^3$ branched over $K$ and induced by $ρ$ bounds a connected cover of $B^4$ branched along such a surface. When $G$ is a dihedral group, we show the obstruction can be computed by evaluating the Seifert form of $K$ on a single curve, a so-called characteristic knot associated to $ρ$. When the dihedral obstruction vanishes, we construct the surface $F$ explicitly.

Extending Quotients of Knot Groups over Surfaces in $B^4$

Abstract

Let be a knot with exterior , and denote by a quotient of its group. We give a sharp obstruction to the existence of a connected, oriented, smooth surface with over whose exterior extends surjectively. Equivalently, we determine whether the cover of branched over and induced by bounds a connected cover of branched along such a surface. When is a dihedral group, we show the obstruction can be computed by evaluating the Seifert form of on a single curve, a so-called characteristic knot associated to . When the dihedral obstruction vanishes, we construct the surface explicitly.

Paper Structure

This paper contains 41 sections, 44 theorems, 212 equations, 3 figures.

Table of Contents

  1. The central question
  2. Motivation
  3. Organization of the paper and summary of results
  4. Based quotients of knot groups and the obstruction space
  5. Based groups and $G$-knots
  6. The space $KG$ and the invariant $\Theta_\rho(K)$
  7. $G$-surfaces and the general existence theorem
  8. Geometric preliminaries on surface exteriors
  9. The $G$-Surface Theorem
  10. Taut groups and consequences
  11. Dihedral machinery
  12. Properties of dihedral groups.
  13. Dihedral knots
  14. Cappell-Shaneson characteristic knots
  15. The lifted invariant $\Theta_\rho(K)$ in $\pi_3(K\mathbb{D}_{(2)})$
  16. ...and 26 more sections

Key Result

Theorem 1.1

Let $K\subseteq S^3$ be a knot equipped with a surjection $\rho\colon\pi_1(E_K)\twoheadrightarrow D_n$. Let $S$ be a Seifert surface for $K$, $V$ a Seifert matrix determined by $S$, and $\beta\subseteq\mathring{S}$ a mod $n$ characteristic knot $($Definition def:chark$)$ corresponding to $\rho$. The $\blacktriangleleft$$\blacktriangleleft$

Figures (3)

  • Figure 1: $\Sigma_2 K$ denotes the $2$-fold cover of $S^3$ branched along the knot $K$. The $2$-disk $\widetilde{B}^2$ is the $2$-fold cover of $B^2$ branched along $\{0\}$ and $\widetilde{S}=\partial \widetilde{B}^2$.
  • Figure 2: Constructing a characteristic surface.
  • Figure 3: The Poindexter Knot, $9_{37}$

Theorems & Definitions (126)

  • Theorem 1.1
  • Remark
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Definition 2.1
  • Example 2.2
  • Example 2.3
  • Definition 2.4
  • Example 2.5
  • ...and 116 more