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$\mathbb{Z}$-disks in $\mathbb{C} P^2$

Anthony Conway, Irving Dai, Maggie Miller

Abstract

We study locally flat disks in $(\mathbb{C} P^2)^\circ:=(\mathbb{C} P^2)\setminus \mathring{B^4}$ with boundary a fixed knot $K$ and whose complement has fundamental group $\mathbb{Z}$. We show that up to topological isotopy rel. boundary, such disks necessarily arise by performing a positive crossing change on $K$ to an Alexander polynomial one knot and capping off with a $\mathbb{Z}$-disk in $D^4.$ Such a crossing change determines a loop in $S^3 \setminus K$ and we prove that the homology class of its lift to the infinite cyclic cover leads to a complete invariant of the disk. We prove that this determines a bijection between the set of rel. boundary topological isotopy classes of $\mathbb{Z}$-disks with boundary $K$ and a quotient of the set of unitary units of the ring $\mathbb{Z}[t^{\pm 1}]/(Δ_K)$. Number-theoretic considerations allow us to deduce that a knot $K \subset S^3$ with quadratic Alexander polynomial bounds $0,1,2,4$, or infinitely many $\mathbb{Z}$-disks in $(\mathbb{C} P^2)^\circ$. This leads to the first examples of knots bounding infinitely many topologically distinct disks whose exteriors have the same fundamental group and equivariant intersection form. Finally we give several examples where these disks are realized smoothly.

$\mathbb{Z}$-disks in $\mathbb{C} P^2$

Abstract

We study locally flat disks in with boundary a fixed knot and whose complement has fundamental group . We show that up to topological isotopy rel. boundary, such disks necessarily arise by performing a positive crossing change on to an Alexander polynomial one knot and capping off with a -disk in Such a crossing change determines a loop in and we prove that the homology class of its lift to the infinite cyclic cover leads to a complete invariant of the disk. We prove that this determines a bijection between the set of rel. boundary topological isotopy classes of -disks with boundary and a quotient of the set of unitary units of the ring . Number-theoretic considerations allow us to deduce that a knot with quadratic Alexander polynomial bounds , or infinitely many -disks in . This leads to the first examples of knots bounding infinitely many topologically distinct disks whose exteriors have the same fundamental group and equivariant intersection form. Finally we give several examples where these disks are realized smoothly.
Paper Structure (28 sections, 40 theorems, 161 equations, 9 figures)

This paper contains 28 sections, 40 theorems, 161 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Existence of $\mathbb{Z}$-disks in $(\mathbb{C}P^2)^\circ$.
  3. Classification of $\mathbb{Z}$-disks in $(\mathbb{C}P^2)^\circ$
  4. Knots with quadratic Alexander polynomial: an explicit enumeration
  5. The underlying number theory
  6. Challenges in the smooth category
  7. Further context and related work
  8. Background material
  9. Twisted homology
  10. The Blanchfield form
  11. The equivariant intersection form
  12. Existence of $\mathbb{Z}$-disks in $(\mathbb{C}P^2)^\circ$
  13. Crossing change $\mathbb{Z}$-disks
  14. The definition of generalized crossing change $\mathbb{Z}$-disks
  15. The exterior of a generalized crossing change $\mathbb{Z}$-disk.
  16. ...and 13 more sections

Key Result

Theorem 1.1

Given a knot $K$, the following assertions are equivalent:

Figures (9)

  • Figure 1: Top: a positive crossing change from $K$ to $K'$ realized as a (generalized) crossing change about a curve $\gamma$. Bottom: A generalized positive crossing change about $\gamma$ transforms $K$ into $K'$. In this example, $\gamma$ links $K$ geometrically more than two times (although as always links $K$ zero times algebraically).
  • Figure 2: Left: The twist knot $K_n$ where the box denotes $n$ full twists. Middle: The figure eight knot $K_1$. Right: The right-handed trefoil $K_{-1}$.
  • Figure 3: A positive crossing change curve for $K_{-1}$. By Observation \ref{['obsv:smooth']}, the resulting disk in $(\mathbb{C}P^2)^\circ$ can be taken to be smoothly embedded.
  • Figure 4: From left to right, top to bottom, we illustrate an isotopy from $\gamma$ to $-\gamma$ in the complement of $K_{-1}$.
  • Figure 5: An unoriented unknotting curve for $K_1$. The two possible choices of orientation yield the two isotopy rel. boundary classes of $\mathbb{Z}$-disk for $K_1$ in $(\mathbb{C}P^2)^\circ$.
  • ...and 4 more figures

Theorems & Definitions (110)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Example 1.5
  • Example 1.6
  • Theorem 1.7
  • Remark 1.8
  • Remark 2.1
  • Remark 2.2
  • ...and 100 more