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Taut foliations from knot diagrams

Diego Santoro

Abstract

We prove that if a knot $K$ has a particular type of diagram then all non-trivial surgeries on $K$ contain a coorientable taut foliation. Knots admitting such diagrams include many two-bridge knots, many pretzel knots, many Montesinos knots and more generally all arborescent knots defined by weighted planar trees with more than one vertex such that $1)$ all weights have absolute value greater than one, and $2)$ there exists a weight with absolute value greater than two. The ideas involved in the proof can also be adapted to study surgeries on links and as an application we show that for all surgeries $M$ on the Borromean link it holds that $M$ is not an $L$-space if and only if $M$ contains a coorientable taut foliation.

Taut foliations from knot diagrams

Abstract

We prove that if a knot has a particular type of diagram then all non-trivial surgeries on contain a coorientable taut foliation. Knots admitting such diagrams include many two-bridge knots, many pretzel knots, many Montesinos knots and more generally all arborescent knots defined by weighted planar trees with more than one vertex such that all weights have absolute value greater than one, and there exists a weight with absolute value greater than two. The ideas involved in the proof can also be adapted to study surgeries on links and as an application we show that for all surgeries on the Borromean link it holds that is not an -space if and only if contains a coorientable taut foliation.
Paper Structure (18 sections, 28 theorems, 5 equations, 41 figures)

This paper contains 18 sections, 28 theorems, 5 equations, 41 figures.

Table of Contents

  1. Introduction
  2. Foliations on surgeries on the Borromean link.
  3. Applications to arborescent knots and some braid closures.
  4. Background on branched surfaces and sutured manifolds
  5. Branched surfaces
  6. Sutured manifolds
  7. Proof of the main theorem
  8. Constructing the branched surface $\overline{B}$
  9. Coorientations of the regions in $S^2$.
  10. Coorientations of the discs: local analysis
  11. Coorientations on the torus T
  12. Proof of the theorem when the set $\mathcal{S}$ has more than $4$ elements
  13. Proof in the case when there are at most $4$ elements in $\mathcal{S}$.
  14. Borromean rings and Bing doubles
  15. Relations with Tait graphs
  16. ...and 3 more sections

Key Result

Theorem 1

Let $K$ be a knot in $S^3$ with a diagram $D$ and suppose that $D$ is not isotopic in $S^2$ to one of the diagrams $D_k$ in Figure fig: D_k new or their mirrors. Suppose that: Then $K$ is persistently foliar.

Figures (41)

  • Figure 3: The diagram $D_k$.
  • Figure 4: How to modify the diagram when there is a cycle containing exactly two vertices. We coloured the front and the back of the letter blue and red respectively. When $b$ is odd the back is shown, since the diagram has changed by a $\pi$-rotation.
  • Figure 5: When $n>1$ the knot in figure is persistently foliar. When $n=1$ it is an $L$-space knot.
  • Figure 6: One of the graphs $\mathcal{G}_g$ and $\mathcal{G}_r$ associated to the diagram in figure is not connected. The pretzel knot $P(-2,3,7)$ is an $L$-space knot and hence cannot be persistently foliar.
  • Figure 7: The Borromean link and the set of surgeries on it that have a coorientable taut foliation (in red) and are $L$-spaces (in blue).
  • ...and 36 more figures

Theorems & Definitions (82)

  • Conjecture : L-space conjecture
  • Definition 1.1
  • Definition 1.2
  • Conjecture : L-space knot conjecture
  • Definition 1.3
  • Example 1.4
  • Theorem : \ref{['thm: main theorem']}
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • ...and 72 more