Twisted torus knots T(p,q,3,s) are tunnel number one
We show that twisted torus knots $T(p,q,3,s)$ are tunnel number one. A short spanning arc connecting two adjacent twisted strands is an unknotting tunnel.
math.GT↗
arXiv subjects
Jung Hoon Lee
Publications and source records attributed to Jung Hoon Lee.
Parity criterion and Dehn twists for unstabilized Heegaard splittings
We give a parity condition of a Heegaard diagram to show that it is unstabilized. This improves the result of [5]. As an application, we construct unstabilized Heegaard splittings by Dehn twists on any given Heegaard splitting.
math.GT↗
Parity condition for irreducibility of Heegaard splittings
Casson and Gordon gave the rectangle condition for strong irreducibility of Heegaard splittings [1]. We give a parity condition for irreducibility of Heegaard splittings of irreducible manifolds. As an application, we give examples of non-stabilized Heegaard splittings by doing a single Dehn twist.
math.GT↗
(Disk, Essential surface) pairs of Heegaard splittings that intersect in one point
We consider a Heegaard splitting M=H_1 \cup_S H_2 of a 3-manifold M having an essential disk D in H_1 and an essential surface F in H_2 with |D \cap F|=1. (We require that boundary of F is in S when H_2 is a compressionbody with non-empty "minus" boundary.) Let F be a genus g surface with n boundary components. From S, we obtain a genus g(S)+2g+n-2 Heegaard splitting M=H'_1 \cup_S' H'_2 by cutting H_2 along F and attaching F \times [0,1] to H_1. As an application, by using a theorem due to Casson and Gordon, we give examples of 3-manifolds having two Heegaard splittings of distinct genera where one of the two Heegaard splittings is a strongly irreducible non-minimal genus splitting and it is obtained from the other by the above construction.
math.GT↗
On the union stabilization of two Heegaard splittings
Let two Heegaard splittings $V_1 \cup W_1$ and $V_2 \cup W_2$ of a 3-manifold $M$ be given. We consider the union stabilization $M=V \cup W$ which is a common stabilization of $V_1 \cup W_1$ and $V_2 \cup W_2$ having the property that $V=V_1 \cup V_2$. We show that any two Heegaard splittings of a 3-manifold have a union stabilization. We also give some examples with numerical bounds on the minimal genus of union stabilization. On the other hand, we give an example of a candidate for which the minimal genus of union stabilization is strictly larger than the usual stable genus -- the minimal genus of common stabilization.
math.GT↗