Module description
Syllabus
This module is designed to build on the level 5 modules Metric Spaces and Topology (226) and Discrete Mathematics (251), and sit either side of Geometry of Surfaces (223).
1. Review of topological spaces and their quotients. Combinatorial description of knots, Reidemeister moves, knot invariants and polynomials.
2. Triangulation, construction, and classification of surfaces, Orientation, genus, Euler number. Connected sums. Knots as boundaries.
3. Review of group theory, relations and generators. Homotopy of paths, fundamental group, Van Kampen’s theorem, applications. Covering maps.
Assessment details
Written Examination.
Semester 1 only students will be set an alternative assessment in lieu of in-person exams in January.
Full year students will complete the standard assessment.
Educational aims & objectives
- To develop a coherent exposition of geometrical concepts linking to existing modules at levels 5 and 6, and of value to both further study and employment.
- Specifically, to apply the notion of topological space to study knots, abstract surfaces and paths.
- Thereby, to focus on topics that occur widely in contemporary mathematics, and to illustrate the interaction between algebra, analysis and geometry.
Teaching pattern
Three hours of lectures and one hour of tutorial per week throughout the term
Suggested reading list
Indicative reading list - link to Leganto system where you can search with module code for lists