{"id":{"repo_id":"wku-diss","oai_identifier":"oai:digitalcommons.wku.edu:theses-1081"},"canonical_url":"https://search.dev.ndltd.org/etd/wku-diss/oai:digitalcommons.wku.edu:theses-1081","repository":{"repo_id":"wku-diss","name":"Western Kentucky University","base_url":"https://digitalcommons.wku.edu/do/oai/"},"display":{"title":"On the Breadth of the Jones Polynomial for Certain Classes of Knots and Links","abstract":"The problem of finding the crossing number of an arbitrary knot or link is a hard problem in general. Only for very special classes of knots and links can we solve this problem. Often we can only hope to find a lower bound on the crossing number Cr(K) of a knot or a link K by computing the Jones polynomial of K, V(K). The crossing number Cr(K) is bounded from below by the difference between the greatest degree and the smallest degree of the polynomial V(K). However the computation of the Jones polynomial of an arbitrary knot or link is also difficult in general. The goal of this thesis is to find closed formulas for the smallest and largest exponents of the Jones polynomial for certain classes of knots and links. This allows us to find a lower bound on the crossing number for these knots and links very quickly. These formulas for the smallest and largest exponents of the Jones polynomial are constructed from special rational tangles expansions and using these formulas, we can extend these results to for [sic] special cases of Montesinos knots and links.","abstract_html":"The problem of finding the crossing number of an arbitrary knot or link is a hard problem in general. Only for very special classes of knots and links can we solve this problem. Often we can only hope to find a lower bound on the crossing number Cr(K) of a knot or a link K by computing the Jones polynomial of K, V(K). The crossing number Cr(K) is bounded from below by the difference between the greatest degree and the smallest degree of the polynomial V(K). However the computation of the Jones polynomial of an arbitrary knot or link is also difficult in general. The goal of this thesis is to find closed formulas for the smallest and largest exponents of the Jones polynomial for certain classes of knots and links. This allows us to find a lower bound on the crossing number for these knots and links very quickly. These formulas for the smallest and largest exponents of the Jones polynomial are constructed from special rational tangles expansions and using these formulas, we can extend these results to for [sic] special cases of Montesinos knots and links.","abstract_has_math":false,"creators":["Lorton, Cody"],"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. Claus Ernst (Director), Dr. Mark P. Robinson,Dr. Uta Ziegler"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2009,"date_issued":"2009-05-01T07:00:00Z","date_published":"2009-05-01T07:00:00Z","updated_at":"2026-07-24T06:07:03Z","subjects":["knot theory","Montesinos knots","Jones polynomial","Mathematics","Number Theory"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://digitalcommons.wku.edu/theses/86","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Dr. Claus Ernst (Director), Dr. Mark P. Robinson,Dr. Uta Ziegler"]},{"key":"dc:creator","label":"Author","values":["Lorton, Cody"]}]},{"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","Montesinos knots","Jones polynomial","Mathematics","Number Theory"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://digitalcommons.wku.edu/theses/86"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["The problem of finding the crossing number of an arbitrary knot or link is a hard problem in general. Only for very special classes of knots and links can we solve this problem. Often we can only hope to find a lower bound on the crossing number Cr(K) of a knot or a link K by computing the Jones polynomial of K, V(K). The crossing number Cr(K) is bounded from below by the difference between the greatest degree and the smallest degree of the polynomial V(K). However the computation of the Jones polynomial of an arbitrary knot or link is also difficult in general. The goal of this thesis is to find closed formulas for the smallest and largest exponents of the Jones polynomial for certain classes of knots and links. This allows us to find a lower bound on the crossing number for these knots and links very quickly. These formulas for the smallest and largest exponents of the Jones polynomial are constructed from special rational tangles expansions and using these formulas, we can extend these results to for [sic] special cases of Montesinos knots and links."]},{"key":"dc:title","label":"Title","values":["On the Breadth of the Jones Polynomial for Certain Classes of Knots and Links"]}]}],"canonical_facts":{"dc:contributor":["Dr. Claus Ernst (Director), Dr. Mark P. Robinson,Dr. Uta Ziegler"],"dc:creator":["Lorton, Cody"],"dc:description.abstract":["The problem of finding the crossing number of an arbitrary knot or link is a hard problem in general. Only for very special classes of knots and links can we solve this problem. Often we can only hope to find a lower bound on the crossing number Cr(K) of a knot or a link K by computing the Jones polynomial of K, V(K). The crossing number Cr(K) is bounded from below by the difference between the greatest degree and the smallest degree of the polynomial V(K). However the computation of the Jones polynomial of an arbitrary knot or link is also difficult in general. The goal of this thesis is to find closed formulas for the smallest and largest exponents of the Jones polynomial for certain classes of knots and links. This allows us to find a lower bound on the crossing number for these knots and links very quickly. These formulas for the smallest and largest exponents of the Jones polynomial are constructed from special rational tangles expansions and using these formulas, we can extend these results to for [sic] special cases of Montesinos knots and links."],"dc:identifier":["https://digitalcommons.wku.edu/theses/86"],"dc:subject":["knot theory","Montesinos knots","Jones polynomial","Mathematics","Number Theory"],"dc:title":["On the Breadth of the Jones Polynomial for Certain Classes of Knots and Links"],"dc:type":["Thesis"],"thesis:degree_discipline":["Department of Mathematics and Computer Science"],"thesis:degree_name":["Master of Science"]},"updated_at":"2026-07-24T06:07:03Z"}