2013/10/31 by Corrin Clarkson
Computer Science · Mathematics · #Embedding #Floer homology #Geometric and Algebraic Topology #Heegaard splitting #Homology (biology) #Homotopy and Cohomology in Algebraic Topology #Morse homology #Rank (graph theory) #Topological and Geometric Data Analysis #math.GT #msc:57M27 #msc:57M60
paper · pdf · doi:10.2140/agt.2017.17.1
published as Algebr. Geom. Topol. 17 (2017) 1-16 · 18 pages, added total rank result, restructured exposition to emphasize similarities between the two results
arxiv created 2013/12/17 · openalex created_date 2016/06/24 · openalex publication_date 2017/01/26 · arxiv updated 2017/02/08 · openalex updated_date 2026/08/05
Given an orientation-preserving self-diffeomorphism [math] of a closed, orientable surface [math] with genus at least two and an embedding [math] of [math] into a three-manifold [math] , we construct a mutant manifold by cutting [math] along [math] and regluing by [math] . We will consider whether there exist nontrivial gluings such that for any embedding, the manifold [math] and its mutant have isomorphic Heegaard Floer homology. In particular, we will demonstrate that if [math] is not isotopic to the identity map, then there exists an embedding of [math] into a three-manifold [math] such that the rank of the nontorsion summands of [math] of [math] differs from that of its mutant. We will also show that if the gluing map is isotopic to neither the identity nor the genus-two hyperelliptic involution, then there exists an embedding of [math] into a three-manifold [math] such that the total rank of [math] of [math] differs from that of its mutant.