arXiv subjects

Kate O'Connor

Publications and source records attributed to Kate O'Connor.

4 recordsLinked to original sources

arXiv2026

Weights of essential surfaces in 2-bridge knot complements

Understanding ideal points in the character varieties of knot complements has led to a number of important invariants for 3-manifolds. Ohtsuki (1994) counted the ideal points for character varieties of 2-bridge knot complements, and he made his techniques more concrete in an ensuing paper (1996). Drawing on these ideas, for all 2-bridge knots $K$, we explicitly determine the structure of a Serre tree for each essential surface in the knot complement directly from the knot diagram. Using these trees, we derive a formula for the number of ideal points associated to each incompressible surface.

math.GT↗

arXiv2019

Turaev Hyperbolicity of Classical and Virtual Knots

By work of W. Thurston, knots and links in the 3-sphere are known to either be torus links, or to contain an essential torus in their complement, or to be hyperbolic, in which case a unique hyperbolic volume can be calculated for their complement. We employ a construction of Turaev to associate a family of hyperbolic 3-manifolds of finite volume to any classical or virtual link, even if non-hyperbolic. These are in turn used to define the Turaev volume of a link, which is the minimal volume among all the hyperbolic 3-manifolds associated via this Turaev construction. In the case of a classical link, we can also define the classical Turaev volume, which is the minimal volume among all the hyperbolic 3-manifolds associated via this Turaev construction for the classical projections only. We then investigate these new invariants.

math.GT↗

arXiv2019

TG-Hyperbolicity of Virtual Links

We extend the theory of hyperbolicity of links in the 3-sphere to tg-hyperbolicity of virtual links, using the fact that the theory of virtual links can be translated into the theory of links living in closed orientable thickened surfaces. When the boundary surfaces are taken to be totally geodesic, we obtain a tg-hyperbolic structure with a unique associated volume. We prove that all virtual alternating links are tg-hyperbolic. We further extend tg-hyperbolicity to several classes of non-alternating virtual links. We then consider bounds on volumes of virtual links and include a table for volumes of the 116 nontrivial virtual knots of four or fewer crossings, all of which, with the exception of the trefoil knot, turn out to be tg-hyperbolic.

math.GT↗

arXiv2017

Numbers Represented by a Finite Set of Binary Quadratic Forms

Every quadratic form represents 0; therefore, if we take any number of quadratic forms and ask which integers are simultaneously represented by all members of the collection, we are guaranteed a nonempty set. But when is that set more than just the "trivial" 0? We address this question in the case of integral, positive- definite, reduced, binary quadratic forms. For forms of the same discriminant, we can use the structure of the underlying class group. If, however, the forms have different discriminants, we must apply class field theory.

math.NT↗