Instanton Homology and Fibered Knots: 2-Torsion and Alexander Polynomial
Analysis
This paper investigates the properties of instanton homology, a powerful tool in 3-manifold topology, focusing on its behavior in the presence of fibered knots. The main result establishes the existence of 2-torsion in the instanton homology of fibered knots (excluding a specific case), providing new insights into the structure of these objects. The paper also connects instanton homology to the Alexander polynomial and Heegaard Floer theory, highlighting its relevance to other areas of knot theory and 3-manifold topology. The technical approach involves sutured instanton theory, allowing for comparisons between different coefficient fields.
Key Takeaways
- •Establishes the presence of 2-torsion in the instanton homology of fibered knots.
- •Provides a formula for calculating instanton homology via sutured instanton theory.
- •Connects instanton homology to the Alexander polynomial for knots admitting lens space surgeries.
- •Shows a non-vanishing result for the next-to-top Alexander grading summand of instanton knot homology for unknotting number one knots.
- •Discusses the relation to Heegaard Floer theory.
“The paper proves that the unreduced singular instanton homology has 2-torsion for any null-homologous fibered knot (except for a specific case) and provides a formula for calculating it.”