The focus of this project is the 3-d index, a topological invariant for knots. Although computations can be done on the 3-d index, it is surprisingly difficult to prove identities arising from such knot invariants. This project aims to investigate the 3-d index, including numerical verifications of some conjectures (specifically, verifying the vanishing of the 3-indices of the 3-sphere and lens spaces). This will involve writing a program to compute the 3-d index, as well as working towards proving some of the combinatorial identities which arise from knot invariants. Alternatively, counter examples which disprove a recently proposed generalisation of the 3-d index to 3-manifolds will also be looked for.
Emma McQuire is an undergraduate student at Monash University, studying a Bachelor of Science Advanced—Research (Honours). She majors in mathematics and physics and is inclined towards pure mathematics but is still in the process of discovering which area she is most interested in. Emma usually does mathematics in her room with her dog (who is always sleeping), but she is looking forward to more in-person opportunities with other people as restrictions ease.