title: A study of knot-graphs

authors: Turner, John Christopher

abstract: This thesis is an account of studies made of knot projections, the tools used being graph-theoretic methods and extensions of them. The basic tool is the adjacency matrix. In fact several kinds of adjacency matrix are defined, for knot diagrams which are either nonoriented or are given orientations which reflect certain topological properties of the knots concerned.
First the properties of knot-graph adjacency matrices are studied. In particular, a matrix equation is derived which is shown to be satisfied by the α- and β-adjacency matrices of all knot-graphs; other matrices appearing in the equation are studied.
Examination of types of walk that can be made on a knot-graph leads to a definition of walk-groups and groupoids on knot-graphs. Relations between the Cayley diagrams of walk-groups and certain other knots which are derived from the diagrams lead to interesting correspondences between knots, both finite and infinite.
An extensive study of the spectra of knot-graph adjacency matrices is presented. Methods for obtaining spectra are given, both for general cases and for certain classes of knots.
The concept of balanced orientations of knot-graphs is introduced; and numbers of rooted directed spanning trees of oriented knot-graphs are studied. It is shown that one type of tree number is a knot invariant, and evidence for the invariance of another one is given. A general inequality relating the two types of number is conjectured. The first tree number has several topological interpretations, and these are discussed. Tables of the numbers are given for knots of orders n = 3, ... ,10; and methods for calculating them for various knot classes are developed.
A vertex deletion theorem relating tree numbers of three knot-graphs is proved. Families of twins which arise from vertex deletions in knot-graphs are defined and studied. It is shown that by repeatedly applying the deletion operation on a given alternating knot-graph, one is led to a certain distribution of ‘twists’, which is given the name ‘twist spectrum’. Twist spectra have many interesting topological properties: they are topological invariants, and their moments are powerful knot discriminants; they distinguish between a knot-graph and its mirror image and hence can be used to investigate amphicheirality. 
The twist spectra for prime alternating 1-links up to order n = 9 and for 2- links up to order n = 8 are tabulated. A general algorithm for computing a twist spectrum is given, and formulae are obtained for twist spectra of members of certain knot-classes.

text: A study of knot-graphs This thesis is an account of studies made of knot projections, the tools used being graph-theoretic methods and extensions of them. The basic tool is the adjacency matrix. In fact several kinds of adjacency matrix are defined, for knot diagrams which are either nonoriented or are given orientations which reflect certain topological properties of the knots concerned.
First the properties of knot-graph adjacency matrices are studied. In particular, a matrix equation is derived which is shown to be satisfied by the α- and β-adjacency matrices of all knot-graphs; other matrices appearing in the equation are studied.
Examination of types of walk that can be made on a knot-graph leads to a definition of walk-groups and groupoids on knot-graphs. Relations between the Cayley diagrams of walk-groups and certain other knots which are derived from the diagrams lead to interesting correspondences between knots, both finite and infinite.
An extensive study of the spectra of knot-graph adjacency matrices is presented. Methods for obtaining spectra are given, both for general cases and for certain classes of knots.
The concept of balanced orientations of knot-graphs is introduced; and numbers of rooted directed spanning trees of oriented knot-graphs are studied. It is shown that one type of tree number is a knot invariant, and evidence for the invariance of another one is given. A general inequality relating the two types of number is conjectured. The first tree number has several topological interpretations, and these are discussed. Tables of the numbers are given for knots of orders n = 3, ... ,10; and methods for calculating them for various knot classes are developed.
A vertex deletion theorem relating tree numbers of three knot-graphs is proved. Families of twins which arise from vertex deletions in knot-graphs are defined and studied. It is shown that by repeatedly applying the deletion operation on a given alternating knot-graph, one is led to a certain distribution of ‘twists’, which is given the name ‘twist spectrum’. Twist spectra have many interesting topological properties: they are topological invariants, and their moments are powerful knot discriminants; they distinguish between a knot-graph and its mirror image and hence can be used to investigate amphicheirality. 
The twist spectra for prime alternating 1-links up to order n = 9 and for 2- links up to order n = 8 are tabulated. A general algorithm for computing a twist spectrum is given, and formulae are obtained for twist spectra of members of certain knot-classes.

year: 1984

title: The Neolithic of Vietnam

authors: Piper, Philip J.

subjects: Vietnam

abstract: The Neolithic in Mainland Southeast Asia is considered to be the period of human prehistory that commenced with the transition from hunting and gathering into mixed economies of foraging and farming. It lasted until the appearance of copper and bronze metallurgy and thus dates between circa 2500 and 1200 BC. In Vietnam, Neolithic sites and cultures are recorded especially around the edges of the Red River delta, along the northern and central coasts, and in the southern river basins between the Đông Nai and lower Mekong. This period of Vietnamese prehistory was characterized by the southward movements of East Asian Neolithic populations from southern China across Mainland Southeast Asia, the introduction of cereal cultivation (rice and foxtail millet), and the management of domesticated pigs and dogs. The Neolithic of Vietnam also reveals material culture relationships extending from southern China to central Thailand. Recent research is also starting to highlight social and cultural diversity during this period, together with the development of significant exchange networks.

text: The Neolithic of Vietnam The Neolithic in Mainland Southeast Asia is considered to be the period of human prehistory that commenced with the transition from hunting and gathering into mixed economies of foraging and farming. It lasted until the appearance of copper and bronze metallurgy and thus dates between circa 2500 and 1200 BC. In Vietnam, Neolithic sites and cultures are recorded especially around the edges of the Red River delta, along the northern and central coasts, and in the southern river basins between the Đông Nai and lower Mekong. This period of Vietnamese prehistory was characterized by the southward movements of East Asian Neolithic populations from southern China across Mainland Southeast Asia, the introduction of cereal cultivation (rice and foxtail millet), and the management of domesticated pigs and dogs. The Neolithic of Vietnam also reveals material culture relationships extending from southern China to central Thailand. Recent research is also starting to highlight social and cultural diversity during this period, together with the development of significant exchange networks.

year: 2022-02-14

title: Maori children : Conceptions of Death and Tangihanga

authors: Jacob, Juanita Emily

subjects: Maori

abstract: Research pertaining to Maori children‟s experiences and perceptions of death and tangihanga is sparse. Much of what is available, relating to children and their experiences with death, particularly death of a loved one, is generalised and stems from Western paradigms of knowledge. In contrast, this study aimed to investigate Maori children‟s experience relating to death and tangi through the eyes of Maori parents. Five areas were explored with Maori parents: (1) childhood experiences of Maori parents relating to death and tangi, (2) parental conceptualisation pertaining to ideas of an afterlife (3) how and when Maori parents talk with children about topics relating to death, tangi and an afterlife (4) how Maori children understand and conceptualise these events, and (5) how these practices will continue on in the future. Findings of this study suggest that Maori children: (a) received abundant support from kinship networks to help them cope with their grief, (b) they were included and involved in all aspects relating to death and tangi, (c) they were encouraged to express emotions openly and without restraint, (d) they understood and comprehended death through personal experience and exposure to tangi, (e) the business of grieving came secondary to the fact that tangi was more like a holiday with family, (f) Maori parents informed and talked to their children about death and tangi, and (g) they often use both Christian ideologies and cultural beliefs to explained death and afterlife to their children. From this study we learn that death was not hidden from children, that parents talked with their children in very open and age relevant ways, and considered their children‟s participation in tangi as an important way to grieve and ensure continuity with kinship networks and support. This study suggests that the challenge now is to ensure that these practices continue to persist between parents and their children, and continue on through generations.

text: Maori children : Conceptions of Death and Tangihanga Research pertaining to Maori children‟s experiences and perceptions of death and tangihanga is sparse. Much of what is available, relating to children and their experiences with death, particularly death of a loved one, is generalised and stems from Western paradigms of knowledge. In contrast, this study aimed to investigate Maori children‟s experience relating to death and tangi through the eyes of Maori parents. Five areas were explored with Maori parents: (1) childhood experiences of Maori parents relating to death and tangi, (2) parental conceptualisation pertaining to ideas of an afterlife (3) how and when Maori parents talk with children about topics relating to death, tangi and an afterlife (4) how Maori children understand and conceptualise these events, and (5) how these practices will continue on in the future. Findings of this study suggest that Maori children: (a) received abundant support from kinship networks to help them cope with their grief, (b) they were included and involved in all aspects relating to death and tangi, (c) they were encouraged to express emotions openly and without restraint, (d) they understood and comprehended death through personal experience and exposure to tangi, (e) the business of grieving came secondary to the fact that tangi was more like a holiday with family, (f) Maori parents informed and talked to their children about death and tangi, and (g) they often use both Christian ideologies and cultural beliefs to explained death and afterlife to their children. From this study we learn that death was not hidden from children, that parents talked with their children in very open and age relevant ways, and considered their children‟s participation in tangi as an important way to grieve and ensure continuity with kinship networks and support. This study suggests that the challenge now is to ensure that these practices continue to persist between parents and their children, and continue on through generations.

year: 2011