论文标题
在两个属中区分波浪和斜率
Distinguished waves and slopes in genus two
论文作者
论文摘要
If R is a nonseparating simple closed curve on the boundary of a genus two handlebody H and H[R] has incompressible boundary, then there exists a unique arc omega in bdry(H), meeting R only in its endpoints, such that, omega is isotopic in bdry(H), keeping its endpoints on R, to a nontrivial wave based at R in each Heegaard diagram of R on bdry(H) which没有切割的vertex。然后，沿R沿其“杰出波浪”欧米茄（Omega）在R上进行手术产生了一对简单的闭合曲线，例如M_1和M_2，在BDRY（H）中，每个曲线都代表BDRY（H [R]）的“ Omega-dendermined-Slope” M，仅取决于R和H。 一些后果：1）在斜率m处仅填充h [r]可以产生s^3，（s^1 x s^2）＃l（p，q）或s^1 x s^2。因此，h [r]嵌入s^3，（s^1 x s^2）＃l（p，q）或s^1 x s^2中的最多嵌入。而且，如果存在这样的嵌入，它是独一无二的。 2）theta曲线是由S^3，s^1 x s^2中的隧道单个结隧道引起的，或（s^1 x s^2）＃l（p，q），具有规范组件结。 3）一个人可以在s^3或s^1 x s^2中识别（1,1）（1,1）结的隧道。 4）识别属的算法S^3，S^1 x s^2的两个Heegaard图，或（S^1 x s^2）＃l（p，q），可以简化使用波的使用。 5）在S^3，s^1 x s^2中计算结节隧道的深度的有效过程，或（s^1 x s^2）＃l（p，q）。 最后，如果H [R_1]对H [R_2]是同型的，但是（H，R_1）和（H，R_2）不是同型的，则BDRY（H [R_1]）上的欧米茄确定的斜率（H [R_1]）和BDRY（H [R_2]）可能有所不同。 However, computation suggests that, if 'R' is a set of simple closed curves on bdry(H) such that R_1 in 'R' and R_2 in 'R' means H[R_1] is homeomorphic to H[R_2], then at most two distinct slopes appear as omega-determined slopes for curves in 'R', and that, if such distinct omega-determined slopes exist, they are never more than distance one除外。
If R is a nonseparating simple closed curve on the boundary of a genus two handlebody H and H[R] has incompressible boundary, then there exists a unique arc omega in bdry(H), meeting R only in its endpoints, such that, omega is isotopic in bdry(H), keeping its endpoints on R, to a nontrivial wave based at R in each Heegaard diagram of R on bdry(H) which has no cut-vertex. Then surgery on R along its "distinguished-wave" omega yields a pair of simple closed curves, say m_1 and m_2, in bdry(H), each representing an "omega-determined-slope" m on bdry(H[R]), that depends only on R and H. A few consequences: 1) Only Dehn filling of H[R] at slope m can yield S^3, (S^1 X S^2) # L(p,q), or S^1 X S^2. So H[R] embeds in at most one of S^3, (S^1 X S^2) # L(p,q), or S^1 X S^2. And, if such an embedding exists, it is unique. 2) Theta curves arising from unknotting tunnels of tunnel-number-one knots in S^3, S^1 X S^2, or (S^1 X S^2) # L(p,q), have canonical component knots. 3) One can recognize (1,1) tunnels of (1,1) knots in S^3 or S^1 X S^2. 4) Algorithms for recognizing genus two Heegaard diagrams of S^3, S^1 X S^2, or (S^1 X S^2) # L(p,q) that use waves can be streamlined. 5) Efficient procedures for computing the depth of an unknotting tunnel of a knot in S^3, S^1 X S^2, or (S^1 X S^2) # L(p,q) exist. Finally, if H[R_1] is homeomorphic to H[R_2], but (H,R_1) and (H,R_2) are not homeomorphic, then the omega-determined slopes on bdry(H[R_1]) and bdry(H[R_2]) may differ. However, computation suggests that, if 'R' is a set of simple closed curves on bdry(H) such that R_1 in 'R' and R_2 in 'R' means H[R_1] is homeomorphic to H[R_2], then at most two distinct slopes appear as omega-determined slopes for curves in 'R', and that, if such distinct omega-determined slopes exist, they are never more than distance one apart.
