The goal of this survey paper is to sum up known results about Heegaard splittings of knot
exteriors in S 3 and present them with some historical perspective. Until the mid 80's …

High distance knots Page 1 Algebraic & Geometric Topology 7 (2007) 1471–1483 1471 High
distance knots YAIR NMinsky YOAV MORIAH SAUL SCHLEIMER We construct knots in S3 …

Let K be a knot in a closed orientable irreducible 3-manifold M and let P be a Heegaard
splitting of the knot complement of genus at least two. Suppose Q is a bridge surface for K …

We prove a theorem that bounds the Heegaard genus from below under special kinds of
toroidal amalgamations of 3–manifolds. As a consequence, we conclude that t (K 1# K 2)≥ …

A Heegaard splitting is a type of combinatorial structure on an orientable compact 3-
manifold. We will give a survey on Heegaard spliting and its applications, including those …

We show that, for any integer $ n\ge 3$, there is a prime knot $ k $ such that (1) $ k $ is not
meridionally primitive, and (2) for every $ m $-bridge knot $ k'$ with $ m\leq n $, the tunnel …

Thin position for knots and for 3–manifolds have become basic tools for 3–manifold
topologists and knot theorists. When David Gabai first introduced the notion of thin position …

The spectrum of the growth rate of the tunnel number is infinite Page 1 PROCEEDINGS OF
THE AMERICAN MATHEMATICAL SOCIETY Volume 144, Number 8, August 2016, Pages …

We show that there exist knots K⊂ S3 with g (E (K))= 2 and g (E (K# K# K))= 6. Together with
[Tsuyoshi Kobayashi, Yo'av Rieck, On the growth rate of the tunnel number of knots, J. Reine …

If the tunnel number of a knot K is denoted t (K), a pair of knots K1, K2 is said to be
subadditive if t (K1)+ t (K2)> t (K1# K2). Scharlemann and Schultens (2000)[11] defined the …