Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Equivariantly slicing strongly negative amphichiral knots

Keegan Boyle, Ahmad Issa

Abstract

We prove obstructions to a strongly negative amphichiral knot bounding an equivariant slice disk in the 4-ball using the determinant, Spinc-structures and Donaldson's theorem. Of the 16 slice strongly negative amphichiral knots with 12 or fewer crossings, our obstructions show that 8 are not equivariantly slice, we exhibit equivariant ribbon diagrams for 5 others, and the remaining 3 are unknown. Finally, we give an obstruction to a knot being strongly negative amphichiral in terms of Heegaard Floer correction terms.

Equivariantly slicing strongly negative amphichiral knots

Abstract

We prove obstructions to a strongly negative amphichiral knot bounding an equivariant slice disk in the 4-ball using the determinant, Spinc-structures and Donaldson's theorem. Of the 16 slice strongly negative amphichiral knots with 12 or fewer crossings, our obstructions show that 8 are not equivariantly slice, we exhibit equivariant ribbon diagrams for 5 others, and the remaining 3 are unknown. Finally, we give an obstruction to a knot being strongly negative amphichiral in terms of Heegaard Floer correction terms.

Paper Structure

This paper contains 8 sections, 15 theorems, 34 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. Lifting the action to the double branched cover
  4. A condition on the determinant
  5. An obstruction on spinc structures
  6. An alternating slice strongly negative amphichiral example
  7. Heegaard Floer correction terms
  8. A table of slice strongly negative amphichiral prime knots with 12 or fewer crossings

Key Result

Theorem \oldthetheorem

If $K$ is an equivariantly slice strongly negative amphichiral knot, then $\hbox{det}(K)$ is the square of a sum of two squares.

Figures (6)

  • Figure 1: A strongly negative amphichiral diagram for $8_9$. The symmetry is given by $\pi$-rotation around an axis perpendicular to the page followed by a reflection across the plane of the diagram. An equivariant slice disk can be seen by performing the band moves shown in red.
  • Figure 2: An oriented edge of $\mathcal{G}(F_+)$ in black intersecting an edge of $\mathcal{G}(F_-)$ in red (left). The orientation on the red edge is induced by the right hand rule. On the right is the oriented arc $\alpha_i'$ induced from the oriented edge of $\mathcal{G}(F_+)$ in black.
  • Figure 3: The arcs $(\alpha_i)_+$ and $(\alpha_i)_-$ are contained in the horizontal and vertical checkerboard surfaces respectively. The green arrow indicates an isotopy between them in $S^3$. Lifting this to $\Sigma(S^3,K)$, we see that the self pairing of the sphere $e_i$ is 1.
  • Figure 4: If $v_i \in \mathcal{G}(F_+)$ is the starting endpoint of an edge corresponding to $e_j$, then $e_j \cdot v_i = 1$. The magenta loop is the boundary of the gray disk $e_j \cap N_+$, and is oriented so that the arc coming out of the page is isotopic (keeping the endpoints on $K$) to $\alpha_j'$ (see Figure \ref{['fig:arcorientation']}) in the complement of $F_+ \cup F_-$.
  • Figure 5: A strongly negative amphichiral symmetry on $12a_{1105}$. The symmetry is $\pi$-rotation within the plane of the diagram followed by a reflection across the plane of the diagram.
  • ...and 1 more figures

Theorems & Definitions (36)

  • Definition \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Conjecture \oldthetheorem
  • Proposition \oldthetheorem
  • proof
  • Corollary \oldthetheorem
  • proof
  • Proposition \oldthetheorem
  • ...and 26 more