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Knots in circle bundles are determined by their complements

Tommaso Cremaschi, Andrew Yarmola

Abstract

We resolve a case of the oriented knot complement conjecture by showing that knots in an orientable circle bundle $N$ over a genus $g \geq 2$ surface $S$ are determined by their complements. We apply this to the setting of canonical knots in the unit and projective tangent bundles, which are knots that are the set of tangents to a closed curve on $S$. We show that canonical knots have homeomorphic complements if and only if their shadows differ by Reidemeister moves, (de)stabilizations, loops/cusps added by transvections, and mapping classes of $S$.

Knots in circle bundles are determined by their complements

Abstract

We resolve a case of the oriented knot complement conjecture by showing that knots in an orientable circle bundle over a genus surface are determined by their complements. We apply this to the setting of canonical knots in the unit and projective tangent bundles, which are knots that are the set of tangents to a closed curve on . We show that canonical knots have homeomorphic complements if and only if their shadows differ by Reidemeister moves, (de)stabilizations, loops/cusps added by transvections, and mapping classes of .
Paper Structure (9 sections, 18 theorems, 17 equations, 7 figures)

This paper contains 9 sections, 18 theorems, 17 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Organization.
  3. Acknowledgements.
  4. Background
  5. Knots in circle bundles and their complements
  6. Application to canonical links
  7. Reidemeister moves for canonical diagrams
  8. Canonical links in tangent bundles
  9. Exteriors of canonical links

Key Result

Theorem 1.1

Let $M$ be a compact orientable 3-manifold where $\partial M = T$ is a torus. Consider slopes $s_1$ and $s_2$ on $T$. Assume that $M(s_1) \cong^+ M(s_2)$ and that $N = M(s_1)$ is an orientable circle bundle over an orientable surface of negative Euler characteristic. Then either $s_1 = s_2$ or both

Figures (7)

  • Figure 1: Schematic of tubing on a whitehead clasp knot in a trivial bundle. The section to which we add a one handle is in blue.
  • Figure 2: A transvection adding a left loop. The right picture has an isotopy applied after a transvection about the grey vertical annulus/torus. The points are there to help visualize the intersections with the vertical annulus/torus.
  • Figure 3: Legendrian Reidemaister moves and (de)stabilizations.
  • Figure 4: Loop Reidemaister moves and (de)stabilizations.
  • Figure 5: On the left, we show $\delta$ and above $A_\delta$ with $\bar{\kappa}$. On the right, we show $\delta_1$ with its canonical lift $\widehat{\delta}_1$. By convention, turning left goes "up" in $UT(S)$.
  • ...and 2 more figures

Theorems & Definitions (35)

  • Theorem 1.1
  • Corollary 1
  • Remark 2
  • Theorem 1.2
  • Corollary 1.2
  • Theorem 1.3
  • Theorem 3.1
  • Remark 3
  • Lemma 4
  • proof : Proof
  • ...and 25 more