Knots and Combinatorics

This project will study a topological invariant of knots called the 3-d index which arose out
of mathematical physics. A first goal for the project would be to write a program (likely in
MATLAB) to compute the 3-d index of some basic knot complements such as the knot
complements of the trefoil and figure-eight knots. The 3-d index has also been generalised
to closed 3-manifolds and so we would look to numerically verify the conjectured vanishing
of the 3-d indices of the 3-sphere and lens spaces. Finally in this project we would also
hopefully prove some combinatorial identities arising from the invariances of the 3-d index.

Oscar Eden

Monash University

Oscar Eden is a third-year undergraduate student at Monash University, studying a Bachelor of Science Advanced – Research. Completing an extended major in pure mathematics and a major in physics, Oscar is interested in physical mathematics, which broadly encompasses the mathematics (which includes topology, geometry, analysis, group theory and Lie algebras) arising from theoretical physics. Outside of his interests in science, Oscar coaches’ tennis and enjoys playing and listening to music.

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