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Balanced algebraic unknotting, linking forms, and surfaces in three- and four-space

Peter Feller, Lukas Lewark

Abstract

We provide three 3-dimensional characterizations of the Z-slice genus of a knot, the minimal genus of a locally-flat surface in 4-space cobounding the knot whose complement has cyclic fundamental group: in terms of balanced algebraic unknotting, in terms of Seifert surfaces, and in terms of presentation matrices of the Blanchfield pairing. This result generalizes to a knot in an integer homology 3-sphere and surfaces in certain simply connected signature zero 4-manifolds cobounding this homology sphere. Using the Blanchfield characterization, we obtain effective lower bounds for the Z-slice genus from the linking pairing of the double branched covering of the knot. In contrast, we show that for odd primes p, the linking pairing on the first homology of the p-fold branched covering is determined up to isometry by the action of the deck transformation group on said first homology. As an application of the new upper and lower bounds, we complete the calculation of the Z-slice genus for all prime knots with crossing number up to 12.

Balanced algebraic unknotting, linking forms, and surfaces in three- and four-space

Abstract

We provide three 3-dimensional characterizations of the Z-slice genus of a knot, the minimal genus of a locally-flat surface in 4-space cobounding the knot whose complement has cyclic fundamental group: in terms of balanced algebraic unknotting, in terms of Seifert surfaces, and in terms of presentation matrices of the Blanchfield pairing. This result generalizes to a knot in an integer homology 3-sphere and surfaces in certain simply connected signature zero 4-manifolds cobounding this homology sphere. Using the Blanchfield characterization, we obtain effective lower bounds for the Z-slice genus from the linking pairing of the double branched covering of the knot. In contrast, we show that for odd primes p, the linking pairing on the first homology of the p-fold branched covering is determined up to isometry by the action of the deck transformation group on said first homology. As an application of the new upper and lower bounds, we complete the calculation of the Z-slice genus for all prime knots with crossing number up to 12.

Paper Structure

This paper contains 33 sections, 47 theorems, 96 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. The genus zero case
  3. Three-dimensional consequences
  4. Four-dimensional consequences
  5. Perspective: The $\mathbb{Z}$--slice genus is a balanced algebraic unknotting number
  6. Proof of \ref{['thm:main']}
  7. Attempts at variations of the main theorem
  8. A generalization of the main theorem to integral homology spheres and $\mathbb{Z}$--slice surfaces in other four-manifolds
  9. Linking forms of cyclic branched coverings
  10. Structure of the paper
  11. Preliminaries
  12. Matrix presentations of pairings on torsion modules
  13. Links, Seifert surfaces, and 3D--cobordism
  14. Twisted homology
  15. Alexander module
  16. ...and 18 more sections

Key Result

Theorem 1.1

For a knot $K$---a smooth, oriented, non-empty, and connected 1--submanifold of $S^3$---and a non-negative integer $g$, the following are equivalent.

Figures (6)

  • Figure 1: The genus-one pretzel knot $(3,3,3)$ (on the left) can be turned into a knot with Alexander polynomial $1$ (on the right) by changing one negative and one positive crossing.
  • Figure 2: Three 4--manifolds, each as union of two others.
  • Figure 3: (a) A balanced crossing change, diagrammatically. (b) Two isotopic drawings of the intersection of ball with a genus 1 Seifert surface realizing a balanced crossing change.
  • Figure 4: Top: the result of performing the two crossing changes in the middle is the same as the result of performing those to the left or right, where left or right depends on the sign of the Dehn surgery along the boundary of the blue (gray-scale: dark) disk. The sign of the Dehn surgery along the boundary of the yellow (gray-scale: light) disk is not relevant. Bottom: the result of performing one of the crossing changes---performing $-1$-framed Dehn surgery (left) and $1$-framed Dehn surgery (right) on the boundary of the blue disk---yields isotopic yellow crossing disks; hence isotopic knots when both crossing changes are performed.
  • Figure 5: Schematic drawing of $F'$ (dimensions reduced by one). 1--, 2--, 3-- and 4--manifolds are drawn in blue (small ellipses), purple (lines), brown (cone levels) and black (cone over large ellipse), respectively.
  • ...and 1 more figures

Theorems & Definitions (86)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3: FreedmanQuinn_90_TopOf4Manifolds; see also GaroufalidisTeichner_04_OnKnotswithtrivialAlex
  • Corollary 1.4
  • proof
  • Corollary 1.5
  • Corollary 1.6
  • Corollary 1.7
  • proof
  • Theorem 1.8: FellerMillerPinzon; see also zbMATH07442350
  • ...and 76 more