232 research outputs found
Recent trends and future directions in vertex-transitive graphs
A graph is said to be vertex-transitive if its automorphism group acts transitively on the vertex set. Some recent developments and possible future directions regarding two famous open problems, asking about existence of Hamilton paths and existence of semiregular automorphisms in vertex-transitive graphs, are discussed, together with some recent results on arc-transitive graphs and half-arc-transitive graphs, two special classes of vertex-transitive graphs that have received particular attention over the last decade
Integration and conjugacy in knot theory
This thesis consists of three self-contained chapters. The first two concern
quantum invariants of links and three manifolds and the third contains results
on the word problem for link groups.
In chapter 1 we relate the tree part of the Aarhus integral to the
mu-invariants of string-links in homology balls thus generalizing results of
Habegger and Masbaum.
There is a folklore result in physics saying that the Feynman integration of
an exponential is itself an exponential. In chapter 2 we state and prove an
exact formulation of this statement in the language which is used in the theory
of finite type invariants.
The final chapter is concerned with properties of link groups. In particular
we study the relationship between known solutions from small cancellation
theory and normal surface theory for the word and conjugacy problems of the
groups of (prime) alternating links. We show that two of the algorithms in the
literature for solving the word problem, each using one of the two approaches,
are the same. Then, by considering small cancellation methods, we give a normal
surface solution to the conjugacy problem of these link groups and characterize
the conjugacy classes. Finally as an application of the small cancellation
properties of link groups we give a new proof that alternating links are
non-trivial.Comment: University of Warwick Ph.D. thesi
Surface quotients of hyperbolic buildings
Let I(p,v) be Bourdon's building, the unique simply-connected 2-complex such
that all 2-cells are regular right-angled hyperbolic p-gons and the link at
each vertex is the complete bipartite graph K(v,v). We investigate and mostly
determine the set of triples (p,v,g) for which there exists a uniform lattice
{\Gamma} in Aut(I(p,v)) such that {\Gamma}\I(p,v) is a compact orientable
surface of genus g. Surprisingly, the existence of {\Gamma} depends upon the
value of v. The remaining cases lead to open questions in tessellations of
surfaces and in number theory. Our construction of {\Gamma}, together with a
theorem of Haglund, implies that for p>=6, every uniform lattice in Aut(I)
contains a surface subgroup. We use elementary group theory, combinatorics,
algebraic topology, and number theory.Comment: 23 pages, 4 figures. Version 2 incorporates referee's suggestions
including new Section 7 discussing relationships between our constructions,
previous examples, and surface subgroups. To appear in Int. Math. Res. No
Geometric Properties of Closed Three Manifolds and Hyperbolic Links
The Geometrization Theorem for 3-manifolds states that every closed orientable 3-manifold can be cut along spheres and tori into pieces which have a geometric structure modeled on one of the eight, 3-dimensional geometries. In joint work with Dennis Sullivan, we combine the different geometries on the toroidal ends of 3-manifolds to describe a uniform geometric structure for all oriented closed prime 3-manifolds. Hyperbolic structures on links in the thickened torus and their geometric properties have been of great interest recently. We discuss geometric properties of augmented and fully augmented links in the thickened torus. We show how sequences of fully augmented links in the 3-sphere which diagrammatically converge to a biperiodic fully augmented link have interesting asymptotic volume growth
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