. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v Chapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Left-orderable groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Left-orderability of 3-manifold groups . . . ...

Finite semisymmetric quasigroups are in bijection with certain mappings between abstract polyhedra and directed graphs, termed alignments. We demonstrate the of any given alignment can always be realized as compact, orientable surfaces. For n to N, class having associated surfaces sum genus ≤ is closed under subobjects homomorphic images. Further, we quasigroup homomorphisms may translated into...

A structure theorem is proven for closed Euclidean 3-dimensional cone manifolds with all cone angles greater than 2π and cone locus a link (no vertices) which allows one to deduce precisely when such a manifold is homotopically atoroidal, and to construct its characteristic submanifold (torus decomposition) when it is not. A by-product of this structure theorem is the result that any Seifert-fi...

We give uniform, explicit, and simple face-pairing descriptions of all the branched cyclic covers of the 3–sphere, branched over two-bridge knots. Our method is to use the bi-twisted face-pairing constructions of Cannon, Floyd, and Parry; these examples show that the bi-twist construction is often efficient and natural. Finally, we give applications to computations of fundamental groups and hom...

Every closed oriented PL 4-manifold is a branched cover of the 4-sphere branched over a PL-surface with finitely many singularities by Piergallini [Topology 34(3):497-508, 1995]. This generalizes a long standing result by Hilden and Montesinos to dimension four. Izmestiev and Joswig [Adv. Geom. 3(2):191-225, 2003] gave a combinatorial equivalent of the Hilden and Montesinos result, constructing...

In this paper, we introduce a sequence of invariants of a knot K in S3 : the knot Floer homology groups ĤFK(Σm(K); K̃, i) of the preimage of K in the m–fold cyclic branched cover over K . We exhibit ĤFK(Σm(K); K̃, i) as the categorification of a well-defined multiple of the Turaev torsion of Σm(K)− K̃ in the case where Σm(K) is a rational homology sphere. In addition, when K is a two-bridge knot, ...