PL-genus of surfaces in homology balls

2023/01/11 by Jennifer Hom, Hom, Jennifer, Matthew Stoffregen +3
Mathematics · Medicine · #57K18 #57Q60 #Botulinum Toxin and Related Neurological Disorders #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2301.04729

openalex publication_date 2023/01/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider manifold-knot pairs (Y,K) where Y is a homology sphere that bounds a homology ball. We show that the minimum genus of a PL surface Σ in a homology ball X such that ∂ (X, Σ) = (Y, K) can be arbitrarily large. Equivalently, the minimum genus of a surface cobordism in a homology cobordism from (Y, K) to any knot in S³ can be arbitrarily large. The proof relies on Heegaard Floer homology.

Related

• A note on PL-disks and rationally slice knots
• The computational complexity of knot genus in a fixed 3-manifold
• Knots in homology spheres which have simple knot Floer homology are trivial
• A note on the rank of Heegaard Floer homology
• A higher-order genus invariant and knot Floer homology