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Stabilization distance bounds from link Floer homology

András Juhász, Ian Zemke

Abstract

We consider the set of connected surfaces in the 4-ball with boundary a fixed knot in the 3-sphere. We define the stabilization distance between two surfaces as the minimal $g$ such that we can get from one to the other using stabilizations and destabilizations through surfaces of genus at most $g$. Similarly, we consider a double point distance between two surfaces of the same genus, which is the minimum over all regular homotopies connecting the two surfaces of the maximal number of double points appearing in the homotopy. To many of the concordance invariants defined using Heegaard Floer homology, we construct an analogous invariant for a pair of surfaces. We show that these give lower bounds on the stabilization distance and the double point distance. We compute our invariants for some pairs of deform-spun slice disks by proving a trace formula on the full infinity knot Floer complex, and by determining the action on knot Floer homology of an automorphism of the connected sum of a knot with itself that swaps the two summands. We use our invariants to find pairs of slice disks with arbitrarily large distance with respect to many of the metrics we consider in this paper. We also answer a slice disk analogue of Problem 1.105 (B) from Kirby's problem list by showing the existence of non-0-cobordant slice disks.

Stabilization distance bounds from link Floer homology

Abstract

We consider the set of connected surfaces in the 4-ball with boundary a fixed knot in the 3-sphere. We define the stabilization distance between two surfaces as the minimal such that we can get from one to the other using stabilizations and destabilizations through surfaces of genus at most . Similarly, we consider a double point distance between two surfaces of the same genus, which is the minimum over all regular homotopies connecting the two surfaces of the maximal number of double points appearing in the homotopy. To many of the concordance invariants defined using Heegaard Floer homology, we construct an analogous invariant for a pair of surfaces. We show that these give lower bounds on the stabilization distance and the double point distance. We compute our invariants for some pairs of deform-spun slice disks by proving a trace formula on the full infinity knot Floer complex, and by determining the action on knot Floer homology of an automorphism of the connected sum of a knot with itself that swaps the two summands. We use our invariants to find pairs of slice disks with arbitrarily large distance with respect to many of the metrics we consider in this paper. We also answer a slice disk analogue of Problem 1.105 (B) from Kirby's problem list by showing the existence of non-0-cobordant slice disks.

Paper Structure

This paper contains 60 sections, 74 theorems, 465 equations, 45 figures.

Table of Contents

  1. Introduction
  2. Metric filtrations on the set of surfaces bounding a knot
  3. Lower bounds from Heegaard Floer homology
  4. Computing the invariants for deform-spun disks using the trace formula
  5. Families with large distance
  6. The stabilization distance of a pair of surfaces
  7. Deform-spun slice disks
  8. Metric filtrations
  9. The stabilization distance
  10. An upper bound on the distance between 1-roll-spun and 1-twist-spun slice disks
  11. Background on the link Floer TQFT
  12. The full link Floer TQFT
  13. Basepoint actions on link Floer homology
  14. Quasi-stabilizations and basepoint moving maps
  15. Cobordism maps for saddles
  16. ...and 45 more sections

Key Result

Theorem 1.1

Let $S$, $S' \in \mathop{\mathrm{Surf}}\nolimits(K)$. Then Furthermore, for every $t \in [0,2]$. Finally, for every $k \in \mathbb{N}$, If $S$ and $S'$ are disks, then Equation eq:Vk-mu-Sing-intro also holds with $\mu_{\mathop{\mathrm{st}}\nolimits}$ in place of $\mu_{\mathop{\mathrm{Sing}}\nolimits}$.

Figures (45)

  • Figure 2.1: A $(3,2)$-stabilization.
  • Figure 2.2: The slice disks $D_{K, t^{\operatorname{wr}(\mathcal{D})}r}$ and $D_{K, t}$ of $-K \# K$. In the top row, we rotate the diagram counterclockwise a full turn in the plane, and consecutive frames differ by a small rotation.
  • Figure 2.3: Attaching a 1-handle to the disk $D_{K, t}$ by adding a pair of bands to the beginning of the movie for $D_{K,t}$.
  • Figure 2.4: On the top row, we show a portion of the surface $S_{\boldsymbol{\mathbf{c}}}$ when $c_i \in \boldsymbol{\mathbf{c}}$. The upper strand of a crossing $c_i$ passes over $\partial B$. On the bottom row, we show a portion of the movie for $S_{\boldsymbol{\mathbf{c}}}$ when $c_i \not\in \boldsymbol{\mathbf{c}}$. The upper strand of the crossing $c_i$ passes underneath $\partial B$.
  • Figure 2.5: A movie for a common stabilization of $S_{\boldsymbol{\mathbf{c}}}$ and $S_{\boldsymbol{\mathbf{c}} \setminus \{c_i, c_j\}}$, when $c_i$ and $c_j$ are consecutive crossings in $\boldsymbol{\mathbf{c}}$ of opposite sign. In the movie, a band is added between the first and the second frames. An isotopy connects the second and third frames. The third and fourth frames are related by attaching the dual band.
  • ...and 40 more figures

Theorems & Definitions (178)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Remark 1.5
  • Definition 2.1
  • Lemma 2.2
  • Definition 2.3
  • Definition 2.4
  • Example 2.5
  • ...and 168 more