2017/01/31 by Robert Myers
Mathematics · #Advanced Combinatorial Mathematics #Epimorphism #Fibered knot #Floer homology #Geometric and Algebraic Topology #Homology (biology) #Knot (papermaking) #Mathematical Dynamics and Fractals #Seifert surface #math.GT #msc:57M50 #msc:57N10
paper · pdf · doi:10.2140/pjm.2019.298.429
published as Pacific J. Math. 298 (2019) 429-444 · 15 pages, 15 figures, Example of Jones polynomial non-invariance under Seifert surface concordance removed, to appear in a separate paper
openalex created_date 2017/05/19 · arxiv created 2018/05/18 · openalex publication_date 2019/03/08 · arxiv updated 2019/04/10 · openalex updated_date 2026/08/05
This paper proves that every oriented non-disk Seifert surface F for a knot K in S3 is smoothly concordant to a Seifert surface F′ for a hyperbolic knot K′ of arbitrarily large volume. This gives a new and simpler proof of the result of Friedl and of Kawauchi that every knot is S-equivalent to a hyperbolic knot of arbitrarily large volume. The construction also gives a new and simpler proof of the result of Silver and Whitten and of Kawauchi that for every knot K there is a hyperbolic knot K′ of arbitrarily large volume and a map of pairs f:(S3,K′)→ (S3,K) which induces an epimorphism on the knot groups. An example is given which shows that knot Floer homology is not an invariant of Seifert surface concordance. The paper also proves that a set of finite volume hyperbolic 3-manifolds with unbounded Haken numbers has unbounded volumes.