Surgeries, sharp 4–manifolds and the Alexanderpolynomial

2014/12/31 by Duncan McCoy · 4 citations
Mathematics · Medicine · #Advanced Combinatorial Mathematics #Alexander polynomial #Bone health and treatments #Bounding overwatch #Combinatorics #Computer science #Geometric and Algebraic Topology #Geometry #Knot (papermaking) #Knot theory #Manifold (fluid mechanics) #Mathematical analysis #Mathematics #Polynomial #Space (punctuation) #Torus #math.GT

paper · pdf · doi:10.2140/agt.2021.21.2649

published in Algebraic & Geometric Topology 21(5), 2649-2676 (Mathematical Sciences Publishers) · 22 pages, various improvements and corrections

arxiv created 2020/08/03 · openalex publication_date 2021/10/31 · arxiv updated 2021/11/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

Work of Ni and Zhang has shown that for the torus knot T_r,s with r>s>1 every surgery slope p/q ≥ (30)/(67)(r²-1)(s²-1) is a characterizing slope. In this paper, we show that this can be lowered to a bound which is linear in rs, namely, p/q≥ (43)/(4)(rs-r-s). The main technical ingredient in this improvement is to show that if Y is an L-space bounding a sharp 4-manifold which is obtained by p/q-surgery on a knot K in S³ and p/q exceeds 4g(K)+4, then the Alexander polynomial of K is uniquely determined by Y and p/q. We also show that if p/q-surgery on K bounds a sharp 4-manifold, then S³_p'/q'(K) bounds a sharp 4-manifold for all p'/q'≥ p/q.

Citations

• An application of gauge theory to four-dimensional topology
• On the Heegaard Floer homology of branched double-covers
• Three dimensional manifolds, Kleinian groups and hyperbolic geometry
• Holomorphic disks and genus bounds
• Deficiency Symmetries of Surgeries in S³