2012/11/19 by Trent Schirmer, Schirmer, Trent
Mathematics · #Advanced Combinatorial Mathematics #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #math.GT
paper · pdf · doi:10.48550/arxiv.1211.4568
25 pages, 3 figures
arxiv created 2012/11/19 · openalex publication_date 2012/11/19 · arxiv updated 2012/11/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We introduce a new technique for finding lower bounds on the Heegaard genus of a 3-manifold obtained by gluing a pair of 3-manifolds together along an incompressible torus or annulus. We deduce a number of inequalities, including one which implies that t(K1# K2)≥ max t(K1),t(K2), where t(-) denotes tunnel number, K1 and K2 are knots in S3, and K1 is m-small. This inequality is best possible. We also provide an interesting collection of examples, similar to a set of examples found by Schultens and Wiedmann, which show that Heegaard genus can stay persistently low under the kinds of gluings we study here.