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Quasi-alternating surgeries on asymmetric L-space knots in the census

Masakazu Teragaito

TL;DR

This work verifies 18 quasi-alternating surgeries on the 9 asymmetric L-space knots in the SnapPy census by a hands-on Montesinos trick, using strongly invertible surgery diagrams and careful twist-tangle manipulations. For each knot, the authors pass from the given surgery descriptions to a strongly invertible position, perform a targeted sequence of twists on selected components, and apply tangle replacements in the quotient to realize the surgery as the double branched cover of a quasi-alternating or Montesinos knot. The results corroborate the computer-generated list and provide an explicit, non-computer-assisted proof that these knots are L-space and admit exactly two quasi-alternating surgeries. Overall, the paper deepens the link between asymmetry, L-space properties, and quasi-alternating surgeries and furnishes concrete diagrammatic and braid-based realizations suitable for further study and reference.

Abstract

In the SnapPy census, there are 9 asymmetric L-space knots. It is known that each of them admits exactly two quasi-alternating surgeries with the aid of a computer. The purpose of this article is to confirm these surgeries by the Montesinos trick.

Quasi-alternating surgeries on asymmetric L-space knots in the census

TL;DR

This work verifies 18 quasi-alternating surgeries on the 9 asymmetric L-space knots in the SnapPy census by a hands-on Montesinos trick, using strongly invertible surgery diagrams and careful twist-tangle manipulations. For each knot, the authors pass from the given surgery descriptions to a strongly invertible position, perform a targeted sequence of twists on selected components, and apply tangle replacements in the quotient to realize the surgery as the double branched cover of a quasi-alternating or Montesinos knot. The results corroborate the computer-generated list and provide an explicit, non-computer-assisted proof that these knots are L-space and admit exactly two quasi-alternating surgeries. Overall, the paper deepens the link between asymmetry, L-space properties, and quasi-alternating surgeries and furnishes concrete diagrammatic and braid-based realizations suitable for further study and reference.

Abstract

In the SnapPy census, there are 9 asymmetric L-space knots. It is known that each of them admits exactly two quasi-alternating surgeries with the aid of a computer. The purpose of this article is to confirm these surgeries by the Montesinos trick.

Paper Structure

This paper contains 28 sections, 36 theorems, 61 equations, 60 figures, 3 tables.

Table of Contents

  1. Introduction
  2. $t12533$
  3. $37$--surgery
  4. $38$--surgery
  5. $t12681$
  6. $61$--surgery
  7. $62$--surgery
  8. $o9\_38928$
  9. $49$--surgery
  10. $50$--surgery
  11. $o9\_39162$
  12. $64$--surgery
  13. $65$--surgery
  14. $o9\_40363$
  15. $82$--surgery
  16. ...and 13 more sections

Key Result

Lemma 2.1

For the surgery diagram $L14n58444(\frac{5}{2},\frac{1}{2},-1)$, performing $\frac{5}{2}$--surgery on $C_0$ and $\frac{1}{2}$--surgery on $C_1$ changes $C_2$ into the mirror image of $t12533$. Hence, the surgery diagram $L14n58444(\frac{5}{2},\frac{1}{2},-1)$ represents $(-37)$--surgery on the mirro

Figures (60)

  • Figure 1: The list of links. The order of components corresponds to SnapPy.
  • Figure 2: For the link diagram $L14n58444(\frac{5}{2},\frac{1}{2},-1)$, do $-2$ twists on $C_1$ to erase it. The box with integer $-1$ indicates $-1$ (left handed) full twist on the strands.
  • Figure 3: $C_2$ is deformed into the closure of a $4$--braid. A box with a negative integer $i$ contains $i$ full twists. A black dot indicates the axis of the closed braid.
  • Figure 4: Left: A strongly invertible position of the link after $+1$ twist on $C_2$ of $L14n58444$. Right: The knot after the tangle replacement. The tangles $A$ and $B$ are the rational tangle $[6,-2]$.
  • Figure 5: A series of deformations from the knot in Fig. \ref{['fig:v3437']} to the mirror of $K12n407$ (Bottom Right).
  • ...and 55 more figures

Theorems & Definitions (73)

  • Lemma 2.1
  • proof
  • Theorem 2.2
  • proof
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Theorem 2.5
  • proof
  • Lemma 3.1
  • ...and 63 more