Abstract. We show the existence of infinitely many prime knots each of which having in their complements meridional essential surfaces with two boundary components and arbitrarily high genus.
In this paper we define alternating Kauffman states of links and we characterize when the induced state surface is a fiber. In addition, we give a different proof of a similar theorem of Futer, Kalfagianni and Purcell on homogeneous states.2010 Mathematics Subject Classification. 57M25, 57M15, 57M50.
We show a combinatorial argument in the diagram of large class of links, including satellite and hyperbolic links, where for each of which the tunnel number is the minimum possible, the number of its components minus one.
We study 2-string free tangle decompositions of knots with tunnel number two.
As an application, we construct infinitely many counter-examples to a
conjecture in the literature stating that the tunnel number of the connected
sum of prime knots doesn't degenerate by more than one.Comment: 40 pages, 28 figures; Version 2: reference added, minor changes in
tex
We study the existence of incompressible embeddings of surfaces into the genus two handlebody. We show that for every compact surface with boundary, orientable or not, there is an incompressible embedding of the surface into the genus two handlebody. In the orientable case the embedding can be either separating or non-separating. We also consider the case in which the genus two handlebody is replaced by an orientable 3manifold with a compressible boundary component of genus greater than or equal to two.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.