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The $ν^+$-equivalence classes of genus one knots

Kouki Sato

Abstract

The $ν^+$-equivalence is an equivalence relation on the knot concordance group. This relation can be seen as a certain stable equivalence on knot Floer complexes $CFK^{\infty}$, and many concordance invariants derived from Heegaard Floer theory are invariant under the equivalence. In this paper, we show that any genus one knot is $ν^+$-equivalent to one of the trefoil, its mirror and the unknot.

The $ν^+$-equivalence classes of genus one knots

Abstract

The -equivalence is an equivalence relation on the knot concordance group. This relation can be seen as a certain stable equivalence on knot Floer complexes , and many concordance invariants derived from Heegaard Floer theory are invariant under the equivalence. In this paper, we show that any genus one knot is -equivalent to one of the trefoil, its mirror and the unknot.

Paper Structure

This paper contains 42 sections, 112 theorems, 208 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Back grounds and the main theorem
  3. The idea of proof: Estimating $\nu^+$-classes
  4. Formal knot complexes and new concordance invariants
  5. Category of formal knot complexes
  6. Poset filtered chain complexes
  7. Formal knot complexes
  8. Definition
  9. Relationship to abstract infinity complex
  10. Basic properties
  11. Commutative monoid structure
  12. The dual of a formal knot complex
  13. Stabilizers
  14. $\nu^+$-invariant
  15. $\nu^+$-equivalence
  16. ...and 27 more sections

Key Result

Theorem 1.1

Two knots $K_1$ and $K_2$ are $\nu^+$-equivalent if and only if we have the following $\mathop{\mathrm{\mathbb{Z}}}\nolimits^2$-filtered chain homotopy equivalence: where $A_1$,$A_2$ are acyclic, i.e., $H_*(A_1)=H_*(A_2)=0$.

Figures (8)

  • Figure 1: A formal knot complex $C^n$ with genus one.
  • Figure 2: A local picture near $f(a) \cup b$
  • Figure 3: A disk $D$ and a surface $F'$
  • Figure 4: The $(m,n)$-twist knot $K_{m,n}$
  • Figure 5: A description of $F$ by a string link
  • ...and 3 more figures

Theorems & Definitions (189)

  • Theorem 1.1: Ho17
  • Theorem 1.2
  • Corollary 1.3
  • Proposition 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Corollary 1.9
  • Theorem 1.10
  • ...and 179 more