Professor Jessica Purcell, Monash University
A mathematical knot is a circle embedded in the 3‐sphere, and two knots are equivalent if they can be smoothly deformed into each other. Mathematicians seek to classify and distinguish knots. One important tool in modern knot theory (and geometric topology more broadly) is hyperbolic geometry. In an appropriate sense, “most” knots have complements in the 3‐sphere that admit hyperbolic geometry. By Mostow‐Prasad rigidity, a hyperbolic structure is a complete knot invariant. This unit will introduce hyperbolic geometry and its role in knot theory and 3‐manifold topology.
Specific topics will likely include: geometric structures, including developing map and holonomy; Thurston’s gluing and completeness equations and their use in computational tools; discrete subgroups of isometries of hyperbolic space and the thick‐thin decomposition; incomplete structures and hyperbolic Dehn filling; and many examples of hyperbolic knots, including twist knots, fully augmented links, 2‐bridge knots, and alternating knots.
While algebra and analysis are strongly recommended, I will try to make the unit accessible to Honours students who are comfortable with mathematical proofs, provided they are willing to learn basics of groups on their own (if they have not taken algebra), or basics of analysis such as completeness (if they have not taken analysis).
The course is more foundational than expert, but for pure mathematics.
Participation in all lectures and tutorials is expected.
For those completing the subject for their own knowledge/interest, evidence of at least 80% attendance at lectures and tutorials is required to receive a certificate of attendance.
This subject will follow the textbook Hyperbolic Knot Theory, by Jessica Purcell, access to the content will be provided during the course.
Take this QUIZ to self-evaluate and get a measure of the key foundational knowledge required.