57th Cascade Topology Seminar

The 57th Cascade Topology Seminar will take place at Seattle University.

The titles and abstracts for this event are:

Ryan Derby-Talbot: Computing Heegaard genus is NP-Hard

One of the most basic measures of the complexity of a 3-manifold is its “Heegaard genus.” The Heegaard genus essentially measures the smallest way that a 3-manifold can be decomposed into two handlebodies. If a 3-manifold happens to be constructed from submanifolds glued together along boundary components, then it is often the case that the Heegaard genus of the manifold can be determined from the Heegaard genera of the submanifolds. However, calculating the overall Heegaard genus this way can be hard –– NP-hard, in fact. In this talk I will introduce the fundamental ideas of Heegaard splittings and incompressible surfaces in 3-manifolds, and show how they can be combined in calculating Heegaard genus. Then I will show how this calculation can be made to mirror questions of satisfiability of certain Boolean formulas, which allows us to consider its computational hardness.

Daniel Heath: Topological Symmetry Groups of the Petersen Graph

We determine all groups which can occur as the orientation preserving topological symmetry group of some embedding of the Petersen graph $P$ in $S^3$. Christian Millichap: Commensurability of hyperbolic knot and link complements. In general, it is a difficult problem to determine if two manifolds are commensurable, i.e., share a common finite sheeted cover. Here, we will examine some combinatorial and geometric approaches to analyzing commensurability classes of hyperbolic knot and link complements. In particular, we will discuss current work done with Worden to show that the only commensurable hyperbolic 2-bridge link complements are the figure-eight knot complement and the $6_{2}^{2}$ link complement. Part of this analysis also results in an interesting corollary: a hyperbolic 2-bridge link complement cannot irregularly cover a hyperbolic 3-manifold.

Marion Campisi: Neighbors of knots in the Gordian graph

We will show that every knot is one crossing change away from a knot of arbitrarily high bridge number and arbitrarily high bridge distance.

Kate Kearney: Knots, Concordance, and Genus

A knot is a circle embedded in a three-sphere. Knots can be the boundary of surfaces in a variety of different contexts (in $S^3$, in $S^3 \times I$, in B^4, considering surfaces as oriented or non-oriented). In this talk we’ll explore how knot concordance and knot genus describe this relationship between knots and surfaces. In particular, we will define several different genus-type invariants of knots, and explore some examples of computations.

Louis Kauffman: TBA

TBA