The Slice-Ribbon Conjecture, posed by Fox in 1966, is a long-standing open conjecture that posits that every slice knot is a ribbon knot. It is known and easily seen that every ribbon knot is a slice knot, implying that the conjecture is really a statement about the equivalence of the two notions of `slice' and `ribbon'. In 2011, Greene and Jabuka showed that the Slice-Ribbon Conjecture holds for the infinite family of odd 3-stranded pretzel knots. In their work, they give a complete characterization of the slice/ribbon knots in that infinite family. This dissertation is motivated by their work and proves that the family of odd 5-stranded pretzel knots satisfiesa weaker version of the Slice-Ribbon Conjecture: All slice odd 5-stranded pretzel knots are mutant ribbon.
The two extra strands in this case add a level of complexity not seen in the 3-stranded case, precisely with respect to mutation. The main result is obtained through use of the knot signature, Donaldson's Diagonalization Theorem from gauge theory, and d-invariants from Heegaard-Floer theory. From each of these tools, a necessary condition for sliceness can be extracted and this thesis shows that for odd 5-stranded pretzel knots that are not mutant ribbon, these conditions are not met and hence do not apply to the Slice-Ribbon Conjecture.
","abstract_html":"<p>The Slice-Ribbon Conjecture, posed by Fox in 1966, is a long-standing open conjecture that posits that every slice knot is a ribbon knot. It is known and easily seen that every ribbon knot is a slice knot, implying that the conjecture is really a statement about the equivalence of the two notions of `slice' and `ribbon'. In 2011, Greene and Jabuka showed that the Slice-Ribbon Conjecture holds for the infinite family of odd 3-stranded pretzel knots. In their work, they give a complete characterization of the slice/ribbon knots in that infinite family. This dissertation is motivated by their work and proves that the family of odd 5-stranded pretzel knots satisfiesa weaker version of the Slice-Ribbon Conjecture: All slice odd 5-stranded pretzel knots are mutant ribbon.</p> <p>The two extra strands in this case add a level of complexity not seen in the 3-stranded case, precisely with respect to mutation. The main result is obtained through use of the knot signature, Donaldson's Diagonalization Theorem from gauge theory, and d-invariants from Heegaard-Floer theory. From each of these tools, a necessary condition for sliceness can be extracted and this thesis shows that for odd 5-stranded pretzel knots that are not mutant ribbon, these conditions are not met and hence do not apply to the Slice-Ribbon Conjecture.</p>","abstract_has_math":false,"creators":["Bryant, Kathryn"],"institution":null,"degree_name":"Doctor of Philosophy (PhD)","degree_level":"Bryn Mawr only","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2016,"date_issued":"2016-01-01T08:00:00Z","date_published":"2016-01-01T08:00:00Z","updated_at":"2026-07-24T01:23:46Z","subjects":["Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://repository.brynmawr.edu/dissertations/144","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Bryant, Kathryn"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Bryn Mawr only"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Doctor of Philosophy (PhD)"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://repository.brynmawr.edu/dissertations/144"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["The Slice-Ribbon Conjecture, posed by Fox in 1966, is a long-standing open conjecture that posits that every slice knot is a ribbon knot. It is known and easily seen that every ribbon knot is a slice knot, implying that the conjecture is really a statement about the equivalence of the two notions of `slice' and `ribbon'. In 2011, Greene and Jabuka showed that the Slice-Ribbon Conjecture holds for the infinite family of odd 3-stranded pretzel knots. In their work, they give a complete characterization of the slice/ribbon knots in that infinite family. This dissertation is motivated by their work and proves that the family of odd 5-stranded pretzel knots satisfiesa weaker version of the Slice-Ribbon Conjecture: All slice odd 5-stranded pretzel knots are mutant ribbon.
The two extra strands in this case add a level of complexity not seen in the 3-stranded case, precisely with respect to mutation. The main result is obtained through use of the knot signature, Donaldson's Diagonalization Theorem from gauge theory, and d-invariants from Heegaard-Floer theory. From each of these tools, a necessary condition for sliceness can be extracted and this thesis shows that for odd 5-stranded pretzel knots that are not mutant ribbon, these conditions are not met and hence do not apply to the Slice-Ribbon Conjecture.
"]},{"key":"dc:title","label":"Title","values":["Slice implies mutant ribbon for odd 5-stranded pretzel knots"]}]}],"canonical_facts":{"dc:creator":["Bryant, Kathryn"],"dc:description.abstract":["The Slice-Ribbon Conjecture, posed by Fox in 1966, is a long-standing open conjecture that posits that every slice knot is a ribbon knot. It is known and easily seen that every ribbon knot is a slice knot, implying that the conjecture is really a statement about the equivalence of the two notions of `slice' and `ribbon'. In 2011, Greene and Jabuka showed that the Slice-Ribbon Conjecture holds for the infinite family of odd 3-stranded pretzel knots. In their work, they give a complete characterization of the slice/ribbon knots in that infinite family. This dissertation is motivated by their work and proves that the family of odd 5-stranded pretzel knots satisfiesa weaker version of the Slice-Ribbon Conjecture: All slice odd 5-stranded pretzel knots are mutant ribbon.
The two extra strands in this case add a level of complexity not seen in the 3-stranded case, precisely with respect to mutation. The main result is obtained through use of the knot signature, Donaldson's Diagonalization Theorem from gauge theory, and d-invariants from Heegaard-Floer theory. From each of these tools, a necessary condition for sliceness can be extracted and this thesis shows that for odd 5-stranded pretzel knots that are not mutant ribbon, these conditions are not met and hence do not apply to the Slice-Ribbon Conjecture.
"],"dc:identifier":["https://repository.brynmawr.edu/dissertations/144"],"dc:subject":["Mathematics"],"dc:title":["Slice implies mutant ribbon for odd 5-stranded pretzel knots"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Bryn Mawr only"],"thesis:degree_name":["Doctor of Philosophy (PhD)"]},"updated_at":"2026-07-24T01:23:46Z"}