Combinatorial methods in Dehn surgery

This is an expository paper, in which we give a summary of some of the joint work of John Luecke and the author on Dehn surgery. We consider the situation where we have two Dehn fillings $M(\alpha)$ and $M(\beta)$ on a given 3-manifold $M$, each containing a surface that is either essential or a Heegaard surface. We show how a combinatorial analysis of the graphs of intersection of the two corresponding punctured surfaces in $M$ enables one to find faces of these graphs which give useful topological information about $M(\alpha)$ and $M(\beta)$, and hence, in certain cases, good upper bounds on the intersection number $\Delta(\alpha, \beta)$ of the two filling slopes

Toroidal and Boundary-Reducing Dehn Fillings

Let M be a simple 3-manifold with a toral boundary component partial_0 M. If Dehn filling M along partial_0 M one way produces a toroidal manifold and Dehn filling M along partial_0 M another way produces a boundary-reducible manifold, then we show that the absolute value of the intersection number on partial_0 M of the two filling slopes is at most two. In the special case that the boundary-reducing filling is actually a solid torus and the intersection number between the filling slopes is two, more is said to describe the toroidal filling.Comment: 12 pp., AmSTeX, 6 figs.[uses epsf], Topology Appl. (to appear

EVALUASI TARIF BUS ANTARKOTA DALAM PROVINSI BERDASARKAN BIAYA OPERASIONAL KENDARAAN TRAYEK SURABAYA-TRENGGALEK

Evaluasi ini dilakukan untuk mengetahui perbandingan tarif berdasarkan Biaya Operasional Kendaraan (BOK) antara PO.Harapan dan PO.Pelita Indah dan juga mencari berapa biaya operasional kendaraan setiap bus per km. Angkutan Antar Kota Dalam Propinsi (AKDP) adalah salah satu pengangkutan penumpang yang dioperasikan secara teratur/terjadwal dan tepat waktu. Bus PO.Harapan dan PO.Pelita Indahmerupakan angkutan AKDP dengan trayek Surabaya - Trenggalek. Jarak Trayek Surabaya-Trenggalek  kira-kira 188 km.Metode yang dilakukan untuk mengevaluasi tarif terhadap Biaya Operasional Kendaraan adalah metode perhitungan biaya pokok bus angkutan umum kelas ekonomi yang diterbitkan oleh Kementerian Perhubungan dimana merupakan data sekunder.Sedangkan data primer ditujukan untuk mendapatkan gambaran biaya/harga komponen-komponen kendaraan dilakukan dengan wawancara langsung dengan pengusaha angkutan.Berdasarkan hasil evaluasi tarif AKDP didapat BOK bus PO.Harapan dengan nilai sebesar Rp 4.887,79 /bus-km dan besarnya tarif BEP sebesar Rp 16.274,74 /pnp. BOK bus PO.Pelita dengan nilai sebesar Rp 4.589,33 /bus-km dan besarnya tarif BEP sebesar Rp 15.539,34 / pnp. Dengan tarif yang berlaku saat ini untuk kedua perusahan tersebut adalah sebesar RP.20.000,00.Kedua pengusaha angkutan mendapatkan keuntungan dimana tarif yang berlaku saat ini wajar untuk menutupi Biaya Operasional Kendaraan

Branched covers of quasipositive links and L-spaces

Let $L$ be a oriented link such that $\Sigma_n(L)$, the $n$-fold cyclic cover of $S^3$ branched over $L$, is an L-space for some $n \geq 2$. We show that if either $L$ is a strongly quasipositive link other than one with Alexander polynomial a multiple of $(t-1)^{2g(L) + (|L|-1)}$, or $L$ is a quasipositive link other than one with Alexander polynomial divisible by $(t-1)^{2g_4(L) + (|L|-1)}$, then there is an integer $n(L)$, determined by the Alexander polynomial of $L$ in the first case and the Alexander polynomial of $L$ and the smooth $4$-genus of $L$, $g_4(L)$, in the second, such that $n \leq n(L)$. If $K$ is a strongly quasipositive knot with monic Alexander polynomial such as an L-space knot, we show that $\Sigma_n(K)$ is not an L-space for $n \geq 6$, and that the Alexander polynomial of $K$ is a non-trivial product of cyclotomic polynomials if $\Sigma_n(K)$ is an L-space for some $n = 2, 3, 4, 5$. Our results allow us to calculate the smooth and topological 4-ball genera of, for instance, quasi-alternating quasipositive links. They also allow us to classify strongly quasipositive alternating links and $3$-strand pretzel links.Comment: 49 pages, 7 figures, minor corrections and improved exposition, accepted for publication by the Journal of Topolog