{"id":{"repo_id":"wku-diss","oai_identifier":"oai:digitalcommons.wku.edu:theses-1164"},"canonical_url":"https://search.dev.ndltd.org/etd/wku-diss/oai:digitalcommons.wku.edu:theses-1164","repository":{"repo_id":"wku-diss","name":"Western Kentucky University","base_url":"https://digitalcommons.wku.edu/do/oai/"},"display":{"title":"An Algorithm to Generate Two-Dimensional Drawings of Conway Algebraic Knots","abstract":"The problem of finding an efficient algorithm to create a two-dimensional embedding of a knot diagram is not an easy one. Typically, knots with a large number of crossings will not nicely generate two-dimensional drawings. This thesis presents an efficient algorithm to generate a knot and to create a nice two-dimensional embedding of the knot. For the purpose of this thesis a drawing is “nice” if the number of tangles in the diagram consisting of half-twists is minimal. More specifically, the algorithm generates prime, alternating Conway algebraic knots in O(<i>n</i>) time where <i>n</i> is the number of crossings in the knot, and it derives a precise representation of the knot’s nice drawing in O(<i>n</i>) time (The rendering of the drawing is not O(<i>n</i>).). <br> Central to the algorithm is a special type of rooted binary tree which represents a distinct prime, alternating Conway algebraic knot. Each leaf in the tree represents a crossing in the knot. The algorithm first generates the tree and then modifies such a tree repeatedly to reduce the number of its leaves while ensuring that the knot type associated with the tree is not modified. The result of the algorithm is a tree (for the knot) with a minimum number of leaves. This minimum tree is the basis of deriving a 4-regular plane map which represents the knot embedding and to finally draw the knot’s diagram.","abstract_html":"The problem of finding an efficient algorithm to create a two-dimensional embedding of a knot diagram is not an easy one. Typically, knots with a large number of crossings will not nicely generate two-dimensional drawings. This thesis presents an efficient algorithm to generate a knot and to create a nice two-dimensional embedding of the knot. For the purpose of this thesis a drawing is “nice” if the number of tangles in the diagram consisting of half-twists is minimal. More specifically, the algorithm generates prime, alternating Conway algebraic knots in O(&lt;i&gt;n&lt;/i&gt;) time where &lt;i&gt;n&lt;/i&gt; is the number of crossings in the knot, and it derives a precise representation of the knot’s nice drawing in O(&lt;i&gt;n&lt;/i&gt;) time (The rendering of the drawing is not O(&lt;i&gt;n&lt;/i&gt;).). &lt;br&gt; Central to the algorithm is a special type of rooted binary tree which represents a distinct prime, alternating Conway algebraic knot. Each leaf in the tree represents a crossing in the knot. The algorithm first generates the tree and then modifies such a tree repeatedly to reduce the number of its leaves while ensuring that the knot type associated with the tree is not modified. The result of the algorithm is a tree (for the knot) with a minimum number of leaves. This minimum tree is the basis of deriving a 4-regular plane map which represents the knot embedding and to finally draw the knot’s diagram.","abstract_has_math":false,"creators":["Tung, Jen-Fu"],"institution":null,"degree_name":"Master of Science","degree_level":null,"degree_discipline":"Department of Mathematics and Computer Science","degree_department":null,"school":null,"contributors":["Dr. Uta Ziegler (Director), Dr. Claus Ernst, Dr. Mustafa Atici"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2010,"date_issued":"2010-05-01T07:00:00Z","date_published":"2010-05-01T07:00:00Z","updated_at":"2026-07-24T06:07:10Z","subjects":["knot theory","Conway algebraic knots","discrete mathematics","Discrete Mathematics and Combinatorics","Mathematics","Numerical Analysis and Computation"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://digitalcommons.wku.edu/theses/163","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Dr. Uta Ziegler (Director), Dr. Claus Ernst, Dr. Mustafa Atici"]},{"key":"dc:creator","label":"Author","values":["Tung, Jen-Fu"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Department of Mathematics and Computer Science"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Master of Science"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["knot theory","Conway algebraic knots","discrete mathematics","Discrete Mathematics and Combinatorics","Mathematics","Numerical Analysis and Computation"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://digitalcommons.wku.edu/theses/163"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["The problem of finding an efficient algorithm to create a two-dimensional embedding of a knot diagram is not an easy one. Typically, knots with a large number of crossings will not nicely generate two-dimensional drawings. This thesis presents an efficient algorithm to generate a knot and to create a nice two-dimensional embedding of the knot. For the purpose of this thesis a drawing is “nice” if the number of tangles in the diagram consisting of half-twists is minimal. More specifically, the algorithm generates prime, alternating Conway algebraic knots in O(<i>n</i>) time where <i>n</i> is the number of crossings in the knot, and it derives a precise representation of the knot’s nice drawing in O(<i>n</i>) time (The rendering of the drawing is not O(<i>n</i>).). <br> Central to the algorithm is a special type of rooted binary tree which represents a distinct prime, alternating Conway algebraic knot. Each leaf in the tree represents a crossing in the knot. The algorithm first generates the tree and then modifies such a tree repeatedly to reduce the number of its leaves while ensuring that the knot type associated with the tree is not modified. The result of the algorithm is a tree (for the knot) with a minimum number of leaves. This minimum tree is the basis of deriving a 4-regular plane map which represents the knot embedding and to finally draw the knot’s diagram."]},{"key":"dc:title","label":"Title","values":["An Algorithm to Generate Two-Dimensional Drawings of Conway Algebraic Knots"]}]}],"canonical_facts":{"dc:contributor":["Dr. Uta Ziegler (Director), Dr. Claus Ernst, Dr. Mustafa Atici"],"dc:creator":["Tung, Jen-Fu"],"dc:description.abstract":["The problem of finding an efficient algorithm to create a two-dimensional embedding of a knot diagram is not an easy one. Typically, knots with a large number of crossings will not nicely generate two-dimensional drawings. This thesis presents an efficient algorithm to generate a knot and to create a nice two-dimensional embedding of the knot. For the purpose of this thesis a drawing is “nice” if the number of tangles in the diagram consisting of half-twists is minimal. More specifically, the algorithm generates prime, alternating Conway algebraic knots in O(<i>n</i>) time where <i>n</i> is the number of crossings in the knot, and it derives a precise representation of the knot’s nice drawing in O(<i>n</i>) time (The rendering of the drawing is not O(<i>n</i>).). <br> Central to the algorithm is a special type of rooted binary tree which represents a distinct prime, alternating Conway algebraic knot. Each leaf in the tree represents a crossing in the knot. The algorithm first generates the tree and then modifies such a tree repeatedly to reduce the number of its leaves while ensuring that the knot type associated with the tree is not modified. The result of the algorithm is a tree (for the knot) with a minimum number of leaves. This minimum tree is the basis of deriving a 4-regular plane map which represents the knot embedding and to finally draw the knot’s diagram."],"dc:identifier":["https://digitalcommons.wku.edu/theses/163"],"dc:subject":["knot theory","Conway algebraic knots","discrete mathematics","Discrete Mathematics and Combinatorics","Mathematics","Numerical Analysis and Computation"],"dc:title":["An Algorithm to Generate Two-Dimensional Drawings of Conway Algebraic Knots"],"dc:type":["Thesis"],"thesis:degree_discipline":["Department of Mathematics and Computer Science"],"thesis:degree_name":["Master of Science"]},"updated_at":"2026-07-24T06:07:10Z"}