Classification of unknotting tunnels for two bridge knots

In this paper, we show that any unknotting tunnel for a two bridge knot is isotopic to either one of known ones. This together with Morimoto-Sakuma's result gives the complete classification of unknotting tunnels for two bridge knots up to isotopies and homeomorphisms.Comment: 32 pages. Published copy, also available at http://www.maths.warwick.ac.uk/gt/GTMon2/paper14.abs.htm

Scharlemann-Thompson untelescoping of Heegaard splittings is finer than Casson-Gordon's

We show that Scharlemann-Thompson untelescoping of Heegaard splittings is finer than Casson-Gordon's by giving concrete examples.Comment: 14 pages, 8 figure

Manifolds admitting both strongly irreducible and weakly reducible minimal genus Heegaard splittings

We construct infinitely many manifolds admitting both strongly irreducible and weakly reducible minimal genus Heegaard splittings. Both closed manifolds and manifolds with boundary tori are constructed.Comment: 12 page

Knots with g(E(K)) = 2 and g(E(K#K#K)) = 6 and Morimoto's Conjecture

We show that there exist knots K in S^3 with g(E(K))=2 and g(E(K#K#K))=6. Together with Theorem~1.5 of [1], this proves existence of counterexamples to Morimoto's Conjecture (Conjecture 1.5 of [2]). This is a special case of arxiv.org/abs/math.GT/0701765 [1] Tsuyoshi Kobayashi and Yo'av Rieck. On the growth rate of the tunnel number of knots. J. Reine Angew. Math., 592:63--78, 2006. [2] Kanji Morimoto. On the super additivity of tunnel number of knots.Math. Ann., 317(3):489--508, 2000.Comment: 6 pages. Final versio

Morimoto's Conjecture for m-small knots

Let $X$ be the exterior of connected sum of knots and $X_i$ the exteriors of the individual knots. In \cite{morimoto1} Morimoto conjectured (originally for $n=2$) that $g(X) < \sigma_{i=1}^n g(X_i)$ if and only if there exists a so-called \em primitive meridian \em in the exterior of the connected sum of a proper subset of the knots. For m-small knots we prove this conjecture and bound the possible degeneration of the Heegaard genus (this bound was previously achieved by Morimoto under a weak assumption \cite{morimoto2}): $$\sigma_{i=1}^n g(X_i) - (n-1) \leq g(X) \leq \sigma_{i=1}^n g(X_i).$$Comment: 17 pages; to appear in the proceedings of the conference "Musubime no topology (Topology of knots) V" held at Waseda University, 16-19 December, 200

Knot exteriors with additive Heegaard genus and Morimoto's Conjecture

Given integers g_i > 1 (i=1,...,n) we prove that there exist infinitely may knots K_i in S^3 so that g(E(K_i)) = g_i and the Heegaard genus of the exterior of the connected sum of K_1,...,K_n is the sum the Heegaard genera of K_1,...,K_n, that is: g(E(K_1#...#K_n)) = g(E(K_1)) +...+ g(E(K_n)). (Here, E() denotes the exterior and g() the Heegaard genus.) Together with Theorem 1.5 of [1], this proves the existence of counterexamples to Morimoto's Conjecture (Conjecture 1.5 of [2]). [1] Tsuyoshi Kobayashi and Yo'av Rieck. On the growth rate of the tunnel number of knots. J. Reine Angew. Math., 592:63--78, 2006. [2] Kanji Morimoto. On the super additivity of tunnel number of knots.Math. Ann., 317(3):489--508, 2000.Comment: 14 page

Heegaard genus of the connected sum of m-small knots

We prove that if $K_1 \subset M_1,...,K_n \subset M_n$ are m-small knots in closed orientable 3-manifolds then the Heegaard genus of E(#_{i=1}^n K_i) is strictly less than the sum of the Heegaard genera of the $E(K_i)$ ($i=1,...,n$) if and only if there exists a proper subset $I$ of $\{1,...,n\}$ so that #_{i \in I} K_i admits a primitive meridian. This generalizes the main result of Morimoto in \cite{morimoto1}.Comment: 34 pages. Final version, to appear in Communications in Analysis and Geometr

Local detection of strongly irreducible Heegaard splittings via knot exteriors

We study the way a strongly irreducible Heegaard surface $\Sigma$ intersects a knot exterior $X$ embedded in a 3-manifold, and show that if $\Sigma \cap \partial X$ consists of simple closed curves which are essential in both $\Sigma$ and $\partial X$, then the intersection $X \cap \Sigma$ consists of meridional annuli only. As an application we show that when considering two Heegaard surfaces that intersect essentially and spinally (cf. Rubinstein and Shcarlemann) any embedded torus in the union of the two bounds a solid torus.Comment: 12 page

Hyperbolic volume and Heegaard distance

We prove (Theorem~1.5) that there exists a constant $\Lambda > 0$ so that if $M$ is a $(\mu,d)$-generic complete hyperbolic 3-manifold of volume \vol[M] < \infty and $\Sigma \subset M$ is a Heegaard surface of genus g(\Sigma) > \Lambda \vol[M], then $d(\Sigma) \leq 2$, where $d(\Sigma)$ denotes the distance of $\Sigma$ as defined by Hempel. The key for the proof of the main result is Theorem~1.8 which is on independent interest. There we prove that if $M$ is a compact 3-manifold that can be triangulated using at most $t$ tetrahedra (possibly with missing or truncated vertices), and $\Sigma$ is a Heegaard surface for $M$ with $g(\Sigma) \geq 76t+26$, then $d(\Sigma) \leq 2$.Comment: 12pages, 3 figure

The growth rate of the tunnel number of m-small knots

In a previous paper the authors defined the growth rate of the tunnel number of knots, an invariant that measures that asymptotic behavior of the tunnel number under connected sum. In this paper we calculate the growth rate of the tunnel number of m-small knots in terms of their bridge indices