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The JSJ-decomposition of the 3-manifold obtained by 0-surgery along a classical pretzel knot of genus one

Nozomu Sekino

Abstract

We consider the JSJ-decomposition of the 3-manifold obtained by 0-surgery along a classical pretzel knot of genus one. We use the classification of exceptional fillings of minimally twisted five-chain links by B. Martelli, C. Petronio and F. Roukema.

The JSJ-decomposition of the 3-manifold obtained by 0-surgery along a classical pretzel knot of genus one

Abstract

We consider the JSJ-decomposition of the 3-manifold obtained by 0-surgery along a classical pretzel knot of genus one. We use the classification of exceptional fillings of minimally twisted five-chain links by B. Martelli, C. Petronio and F. Roukema.
Paper Structure (25 sections, 2 theorems, 7 equations, 19 figures)

This paper contains 25 sections, 2 theorems, 7 equations, 19 figures.

Table of Contents

  1. Introduction
  2. Classical pretzel knots whose genera are less than or equal to one and their $0$-surgeries
  3. Classical pretzel knots whose genera are less than or equal to one
  4. The complement of a classical pretzel knot of genus one and the 3-manifold obtained by $0$-surgery along it
  5. The 3-manifold obtained by cutting $P_{2p+1,2q+1,2r+1}(0)$ along $T'$
  6. The structure of $\overline{P_{2p+1,2q+1,2r+1}(0)\setminus T'}$
  7. Non-hyperbolic cases
  8. The case where $-1\in \{p,q,r\}$
  9. The case where $0\in \{p,q,r\}$
  10. The case where $\{-2,1\}\subset \{p,q,r\}$
  11. The case where $\{p,q,r\}=\{-2,2,2\}$
  12. The case where $\{p,q,r\}=\{-3,-3,1\}$
  13. Hyperbolic cases
  14. The case where $-2, -1, 0, 1\notin \{p,q,r\}$
  15. The case where $1\in \{p,q,r\}$
  16. ...and 10 more sections

Key Result

Theorem 1.1

Let $\mathcal{T}$ be the set of decomposing tori of the JSJ-decomposition of the 3-manifold obtained by $0$-surgery along a pretzel knot $P_{2p+1,2q+1,2r+1}$. Then we have the following.

Figures (19)

  • Figure 1: the standard diagram of $P_{l,m,n}$ and examples of right half twists
  • Figure 2: the twist knot $K_{j}$ and $P_{-1,-1,2j+1}$
  • Figure 3: $P_{2p+1,2q+1,2r+1}$ and examples of right full twists
  • Figure 4: The standard Seifert surface $T$ of genus one for $P_{2p+1,2q+1,2r+1}$
  • Figure 5: Left: the based loops $a$, $b$ and $t$ in $E_{2p+1,2q+1,2r+1}$ Right: loops $x$ and $y$ on $T$
  • ...and 14 more figures

Theorems & Definitions (8)

  • Theorem 1.1
  • Definition 2.1
  • Lemma 2.1
  • proof
  • Definition 2.2
  • proof
  • Remark 3.1
  • Definition 4.1