Four studies in the geometry of biased graphs

This thesis has four parts. The overall theme is that I study the representability of matroids of a biased expansion graph of rank 3, in a full algebraic matroid and in a projective plane, and the orientability of these matroids.; In the second chapter I study embeddability in projective planes. A quasigroup expansion of K3 gives rise to two matroids of rank 3: the full bias matroid and complete lift matroid. I give an algebraic characterization for the representability of these matroids in a projective plane in terms of quasigroups and ternary rings.; The third chapter concerns algebraic representability of a special kind of matroid. The matroid M(n) where n ≥ 2 is a complete lift matroid associated with the cyclic group Zn . Gordon proved that this matroid is algebraically representable if n is a prime number. Lindstrom proved that M(n) is not algebraically representable if n > 2 is an even number, and he conjectured that if n is a composite number then it is not algebraically representable. I prove this conjecture.; I introduce a generalization of the concept of harmonic conjugation from projective geometry and full algebraic matroids to a larger class of matroids called harmonic matroids. I use harmonic conjugation to prove Lindstrom's conjecture in this more general case. I also use harmonic conjugation to construct a Desarguesian projective plane of prime order in harmonic matroids without making use of the axioms of projective geometry. As a particular case, I have a combinatorial construction of a Desarguesian projective plane of prime order in full algebraic matroids. This is chapter 4.; Chapter 5 is about orientable matroids. Ziegler raised the question: for every prime power q, find a minimal non-orientable submatroid of the projective plane over the q-element field. We answer this question. (This is a collaboration with David Forge.) We construct a new family of minimal non-orientable matroids of rank three (these matroids are bias matroids). Some of these matroids embed in Desarguesian projective planes, thus giving an answer to the question of Ziegler.