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Two curious strongly invertible L-space knots

Kenneth L. Baker, Marc Kegel, Duncan McCoy

TL;DR

This work constructs two strongly invertible L-space knots, $K_1=17nh0000014$ and $K_2=17nh0000019$, that violate Watson's proposed characterization by producing $\varkappa(K,\Phi)$ not supported on a single diagonal, despite both being L-space knots with unique strong inversions. It computes $\varkappa$ via tangle fillings and shows that all rational surgeries yield double branched covers of non-thin links, hence no thin surgeries exist for these knots. The paper further analyzes exceptional and symmetry-exceptional slopes, identifying specific non-thin exceptional slopes and finite symmetry-exceptional slopes with explicit branching-set data, and proves that none of these slopes yield thin manifolds. Additionally, it notes that both knots have formal semigroups that are actual semigroups, highlighting a rare structural property among L-space knots. Overall, the results refute a conjectural thin-surgery criterion and deepen understanding of the interaction between Khovanov invariants, strong inversions, and Dehn surgeries on L-space knots.

Abstract

We present two examples of strongly invertible L-space knots whose surgeries are never the double branched cover of a Khovanov thin link in the 3-sphere. Consequently, these knots provide counterexamples to a conjectural characterization of strongly invertible L-space knots due to Watson. We also discuss other exceptional properties of these two knots, for example, these two L-space knots have formal semigroups that are actual semigroups.

Two curious strongly invertible L-space knots

TL;DR

This work constructs two strongly invertible L-space knots, and , that violate Watson's proposed characterization by producing not supported on a single diagonal, despite both being L-space knots with unique strong inversions. It computes via tangle fillings and shows that all rational surgeries yield double branched covers of non-thin links, hence no thin surgeries exist for these knots. The paper further analyzes exceptional and symmetry-exceptional slopes, identifying specific non-thin exceptional slopes and finite symmetry-exceptional slopes with explicit branching-set data, and proves that none of these slopes yield thin manifolds. Additionally, it notes that both knots have formal semigroups that are actual semigroups, highlighting a rare structural property among L-space knots. Overall, the results refute a conjectural thin-surgery criterion and deepen understanding of the interaction between Khovanov invariants, strong inversions, and Dehn surgeries on L-space knots.

Abstract

We present two examples of strongly invertible L-space knots whose surgeries are never the double branched cover of a Khovanov thin link in the 3-sphere. Consequently, these knots provide counterexamples to a conjectural characterization of strongly invertible L-space knots due to Watson. We also discuss other exceptional properties of these two knots, for example, these two L-space knots have formal semigroups that are actual semigroups.
Paper Structure (6 sections, 9 theorems, 10 equations, 4 figures, 2 tables)

This paper contains 6 sections, 9 theorems, 10 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Khovanov homology and the K-invariant.
  3. Counterexamples to Watson's conjecture
  4. L-space knots without thin surgeries
  5. Exceptional slopes
  6. Symmetry-exceptional slopes

Key Result

Theorem 1.2

There exist two strongly invertible L-space knots, $K_1$ and $K_2$ with unique strong inversions $\Phi_1$ and $\Phi_2$, such that $\varkappa(K_i, \Phi_i)$ is supported in two distinct $\delta$-gradings. In particular,In this article, we adopt the convention that the relative $q$-grading and thus als

Figures (4)

  • Figure 1: Strongly invertible diagrams for $K_1$ (left) and $K_2$ (right).
  • Figure 2: Left: The tangle filling of $T$ with slope $0$ yields a $2$-component link with determinant $0$ whose double branched cover is $K(0)$. Right: The tangle filling of $T$ with slope $\infty$ yields an unknot whose double branched cover is $S^3=K(\infty)$.
  • Figure 3: The tangle exteriors $T_1$ of $K_1$ (left) and $T_2$ of $K_2$ (right), framed by the images of the meridians $\mu$ and Seifert longitudes $\lambda$ under the covering map. Above each are the associated rational tangle fillings.
  • Figure 4: The calculation of the $\varkappa$-invariant of $K_1$ and $K_2$.

Theorems & Definitions (20)

  • Conjecture 1.1: Watson_KH_sym2017
  • Theorem 1.2
  • Theorem 1.3
  • Conjecture 1.7
  • Proposition 3.1
  • proof
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • ...and 10 more