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Knot concordance, the point class in instanton homology and Donaldson invariants

Yuhan Lim

Abstract

We define an invariant ${\varphi}$ for knots in the 3-sphere by means of Donaldson invariants and Floer's instanton homology. Some basic properties of this invariant are established and it is shown that ${\varphi}$ coincides with a special case of an invariant defined by Froyshov

Knot concordance, the point class in instanton homology and Donaldson invariants

Abstract

We define an invariant for knots in the 3-sphere by means of Donaldson invariants and Floer's instanton homology. Some basic properties of this invariant are established and it is shown that coincides with a special case of an invariant defined by Froyshov
Paper Structure (20 sections, 16 theorems, 116 equations)

This paper contains 20 sections, 16 theorems, 116 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and definitions
  3. Proof of Theorem \ref{['thrm-one']}
  4. Immersed negative concordance
  5. Proof of Theorem \ref{['thrm-one']}(\ref{['thrm-item1']})
  6. Some results on the instanton homology for $S^1\times\Sigma$
  7. Proof of Theorem \ref{['thrm-one']}(\ref{['thrm-item2']})
  8. Proof of Theorem \ref{['thrm-one']}(\ref{['thrm-item3']})
  9. Proof of lemma.
  10. Proof of Theorem \ref{['thrm-one']}(\ref{['thrm-item4']})
  11. A cobordism map
  12. Completion of proof
  13. Results on connected sums
  14. The Poincare homology sphere
  15. Connected sums (i)
  16. ...and 5 more sections

Key Result

Theorem 1

Let $K$ be a knot in $S^3$. The following properties hold for ${\varphi}(K)$.

Theorems & Definitions (27)

  • Theorem 1
  • Theorem 2
  • Corollary 1: Fy2
  • Example 1
  • Example 2
  • Example 3
  • Theorem 3
  • Definition 1
  • Remark
  • Remark
  • ...and 17 more