Combinatorial aspects of symmetries on groups

An MSc dissertation by Shivani Singh. University of Witwatersrand Faculty of Science, School of Mathematics. August 2016.These symmetries have interesting applications to enumerative combinatorics and to Ramsey theory. The aim of this thesis will be to present some important results in these fields. In particular, we shall enumerate the r-ary symmetric bracelets of length n.LG201

Quasi-trivial Quandles and Biquandles, Cocycle Enhancements and Link-Homotopy of Pretzel links

We investigate some algebraic structures called quasi-trivial quandles and we use them to study link-homotopy of pretzel links. Precisely, a necessary and sufficient condition for a pretzel link with at least two components being trivial under link-homotopy is given. We also generalize the quasi-trivial quandle idea to the case of biquandles and consider enhancement of the quasi-trivial biquandle cocycle counting invariant by quasi-trivial biquandle cocycles, obtaining invariants of link-homotopy type of links analogous to the quasi-trivial quandle cocycle invariants in Ayumu Inoue's article arXiv:1205.5891.Comment: 14 pages. Version 3 includes some corrections and typo fixe

Symmetric colorings of finite groups

A dissertation submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of requirements for the degree of Master of Science. December 2014.Let G be a finite group and let r ∈ N. A coloring of G is any mapping : G −→ {1, 2, 3, ..., r}. Colorings of G, and are equivalent if there exists an element g in G such that (xg−1) = (x) for all x in G. A coloring of a finite group G is called symmetric with respect to an element g in G if (gx−1g) = (x) for all x ∈ G. We derive formulae for computing the number of symmetric colorings and the number of equivalence classes of symmetric colorings for some classes of finite group

Orientable hyperbolic 4-manifolds over the 120-cell

Since there is no hyperbolic Dehn filling theorem for higher dimensions, it is challenging to construct explicit hyperbolic manifolds of small volume in dimension at least four. Here, we build up closed hyperbolic 4-manifolds of volume $\frac{34\pi^2}{3}\cdot 16$ by using the small cover theory. In particular, we classify all of the orientable four-dimensional small covers over the right-angled 120-cell up to homeomorphism; these are all with even intersection forms.Comment: In this version, we corret some drawing typos in figures of adjacent matrices of $X_{45}$, $X_{55}$, $X_{56}$ and $X_{57}$. The author are grateful to Leonardo Ferrari who pointed out the mistake