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Item Congruences in racks and quandles : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, Manawatū Campus, Palmerston North, New Zealand(Massey University, 2022) Burrows, Wayne JohnRacks and quandles are related algebraic structures based on axioms of invertibility and self-distributivity, and in the case of quandles, an additional idempotence axiom — thus every quandle is a rack. They have practical application as the three quandle axioms algebraically encode the Reidemeister moves of knot theory. However, racks and quandles are interesting and worthy of study in their own right and that is what we do here. Congruences are a means of distilling patterns of behaviour within algebraic structures. They allow us to form a quotient that gives us a coarser view of the structure from which we can discern interesting properties. Congruences need to respect the operations in the algebraic structure. Racks, although often defined in terms of only one binary operation, necessarily, as a result of the invertibility axiom, have two binary operations — a primary rack operation and an inverse rack operation. We have a rack in the quotient only when the congruence respects both operations. A congruence that respects both operations we call a rack congruence or a quandle congruence. Congruences defined in terms of only one of the binary operations may not preserve the rack structure in the quotient. This raises the question of whether congruences that respect only one rack operation — half congruences — can exist. We show they can by constructing examples of half congruences that do not induce a rack in the quotient. For weighted average quandles on Q we completely characterise congruences in terms of certain subgroups of Q. Depending on the weight, congruences can exhibit one of three possible behaviours. Weighted average quandles are a special case of the more general Alexander quandle. For Alexander quandles, we characterise when a congruence induces an Alexander quandle in the quotient. In weighted average quandles every congruence comes from a subgroup of Q. In Alexander quandles, there are additional congruences that do not come from a subgroup. We give examples of congruences that exhibit that more complex behaviour.Item A graph theoretic proof that Wada's type seven link invariant is determined by the double branched cover : a thesis presented in partial fulfilment of the requirements for the degree of Master of Science in Mathematics at Massey University, Manawatū, New Zealand(Massey University, 2021) Solomon, Zachary TancredThe fundamental group of a link L is a group-valued link invariant that can be defined by assigning a generator to each arc of a link diagram of L, and introducing a relation between them at each crossing. Wada studied what he called shift representations to look for other crossing relations that might define group-valued link invariants. He found seven shift representations, two of which he noted do not define group-valued link invariants. One of the seven defines an infinite family Gm of invariants that includes the fundamental group as G₁, and these have since been shown to distinguish knots up to reflection for m ≥ 2. Wada showed that three of the remaining four give no new information, leaving just his type seven invariant, which we call W₇. Sakuma showed that the seventh of Wada’s shift representations is isomorphic to the free product of Z and the fundamental group of the double branched cover of L, π₁(L˜₂), that is W₇(L) ∼= π₁(L˜₂) ∗ Z. We will use graph theoretic methods to give a new proof of Sakuma’s result.