Euler characteristic and quadrilaterals of normal surfaces

Let $M$ be a compact 3-manifold with a triangulation $\tau$. We give an inequality relating the Euler characteristic of a surface $F$ normally embedded in $M$ with the number of normal quadrilaterals in $F$. This gives a relation between a topological invariant of the surface and a quantity derived from its combinatorial description. Secondly, we obtain an inequality relating the number of normal triangles and normal quadrilaterals of $F$, that depends on the maximum number of tetrahedrons that share a vertex in $\tau$.Comment: 7 pages, 1 figur

Face pairing graphs and 3-manifold enumeration

The face pairing graph of a 3-manifold triangulation is a 4-valent graph denoting which tetrahedron faces are identified with which others. We present a series of properties that must be satisfied by the face pairing graph of a closed minimal P^2-irreducible triangulation. In addition we present constraints upon the combinatorial structure of such a triangulation that can be deduced from its face pairing graph. These results are then applied to the enumeration of closed minimal P^2-irreducible 3-manifold triangulations, leading to a significant improvement in the performance of the enumeration algorithm. Results are offered for both orientable and non-orientable triangulations.Comment: 30 pages, 57 figures; v2: clarified some passages and generalised the final theorem to the non-orientable case; v3: fixed a flaw in the proof of the conical face lemm

On iterated torus knots and transversal knots

A knot type is exchange reducible if an arbitrary closed n-braid representative can be changed to a closed braid of minimum braid index by a finite sequence of braid isotopies, exchange moves and +/- destabilizations. In the manuscript [J Birman and NC Wrinkle, On transversally simple knots, preprint (1999)] a transversal knot in the standard contact structure for S^3 is defined to be transversally simple if it is characterized up to transversal isotopy by its topological knot type and its self-linking number. Theorem 2 of Birman and Wrinkle [op cit] establishes that exchange reducibility implies transversally simplicity. The main result in this note, establishes that iterated torus knots are exchange reducible. It then follows as a Corollary that iterated torus knots are transversally simple.Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol5/paper21.abs.htm

Corrigendum to "Knot Floer homology detects fibred knots"

We correct a mistake on the citation of JSJ theory in \cite{Ni}. Some arguments in \cite{Ni} are also slightly modified accordingly.Comment: 3 page

Makrofauna Tanah Perkebunan Kelapa Sawit (Elais Guineensis Jacq) Di Lahan Gambut Dengan Pemberian Bahan Organik Pada Tinggi Muka Air Tanah Berbeda

This researchaims to know the species amountd, total individual, calculate population density (K) and relative density (KR) soil macrofauna in oil palm plantation (Elais guineensis Jacq) in peatland by giving organic matter in water level is different.This research are conducted by observation, sampling for soil macrofauna of data is determined by purposive random sampling method, and data soil macrofauna results to analyzed statistic descriptive. The results showed that the amountd species, total individual, population density and relative density of soil macrofauna were higher in water levels of 40-50 cm. The giving organic matter in the soil water level is different in the first month, giving palm fronds and Mucunna bracteata produces the number of species, the total individual, population density, and density of the soil macrofauna relative higher than that of oil palm empty fruit bunches,while in the third month the species, the total individual, population density, and density of the soil makorfauna ralatif higher in the provision of oil palm empty fruit bunches and palm fronds.In water levels same produce the amountd species, total individual, population density, and relative density of soil macrofauna higher ground in water levels of 40-50 cm except amountd species in the third month

Makrofauna Tanah Perkebunan Kelapa Sawit (Elais Guineensis Jacq) Di Lahan Gambut Dengan Pemberian Bahan Organik Pada Tinggi Muka Air Tanah Berbeda

This researchaims to know the species amountd, total individual, calculate population density (K) and relative density (KR) soil macrofauna in oil palm plantation (Elais guineensis Jacq) in peatland by giving organic matter in water level is different.This research are conducted by observation, sampling for soil macrofauna of data is determined by purposive random sampling method, and data soil macrofauna results to analyzed statistic descriptive. The results showed that the amountd species, total individual, population density and relative density of soil macrofauna were higher in water levels of 40-50 cm. The giving organic matter in the soil water level is different in the first month, giving palm fronds and Mucunna bracteata produces the number of species, the total individual, population density, and density of the soil macrofauna relative higher than that of oil palm empty fruit bunches,while in the third month the species, the total individual, population density, and density of the soil makorfauna ralatif higher in the provision of oil palm empty fruit bunches and palm fronds.In water levels same produce the amountd species, total individual, population density, and relative density of soil macrofauna higher ground in water levels of 40-50 cm except amountd species in the third month

Continuous improvement in the context of organizational culture

Thesis (M.S.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering; and, Thesis (M.S.)--Sloan School of Management, 1996.Includes bibliographical references (p. 114-116).by W. Jaco Smit.M.S

Spatial Graphs with Local Knots

It is shown that for any locally knotted edge of a 3-connected graph in $S^3$, there is a ball that contains all of the local knots of that edge and is unique up to an isotopy setwise fixing the graph. This result is applied to the study of topological symmetry groups of graphs embedded in $S^3$.Comment: 20 pages, 3 figures; in v. 2 the proof of Theorem 1 has been clarified, and other minor revisions have been mad

The Computational Complexity of Knot and Link Problems

We consider the problem of deciding whether a polygonal knot in 3-dimensional Euclidean space is unknotted, capable of being continuously deformed without self-intersection so that it lies in a plane. We show that this problem, {\sc unknotting problem} is in {\bf NP}. We also consider the problem, {\sc unknotting problem} of determining whether two or more such polygons can be split, or continuously deformed without self-intersection so that they occupy both sides of a plane without intersecting it. We show that it also is in NP. Finally, we show that the problem of determining the genus of a polygonal knot (a generalization of the problem of determining whether it is unknotted) is in {\bf PSPACE}. We also give exponential worst-case running time bounds for deterministic algorithms to solve each of these problems. These algorithms are based on the use of normal surfaces and decision procedures due to W. Haken, with recent extensions by W. Jaco and J. L. Tollefson.Comment: 32 pages, 1 figur