An n-dimensional k-handlebody is an n-manifold obtained from an n-ball by attaching handles of index up to k, where n ≥ k. We will discuss that for any n ≥ 2k + 1, any n-dimensional k-handlebody is diffeomorphic to the product of a 2k-dimensional k-handlebody and an (n − 2k)-ball. For example, a 2025-dimensional 6-handlebody is the product of an 12-dimensional 6-handlebody and a 2013-ball. We also introduce (n,k)-Kirby diagrams for some n-dimensional k-handlebodies. Here (4,2)-Kirby diagrams correspond to the classical Kirby diagrams for 4-dimensional 2-handlebodies.

In this talk, we study several computational problems related to knots and links. We investigate lower bounds on the computational complexity of theoretical knot theory problems. Unknotting number is one of the most interesting knot invariants, and various research has been done to find unknotting numbers of knots. However, compared to its simple definition, it is generally hard to find the unknotting number of a knot, and it is known for only some knots. There is no algorithm for determining unknotting numbers yet. First, we show that for an arbitrary positive integer n, a non-torus knot exists with the unknotting number n. Second, we show that the computational complexity of the diagrammatic un-knotting number problem is NP-hard. We construct a Karp reduction from 3-SAT to the diagrammatic unknotting number problem. Third, we also prove that the prime sublink problem is NP-hard by making a Karp reduction from the known NP-complete problem, the non-tautology problem.

If M is a hyperbolic 3-manifold fibering over the circle, then the fundamental group of M acts faithfully by homeomorphisms on a circle—the circle at infinity of the universal cover of the fiber—preserving a pair of invariant (stable and unstable) laminations. Many different kinds of dynamical structures including taut foliations and quasigeodesic or pseudo-Anosov flows are known to give rise to universal circles—a circle with a faithful action of the fundamental group preserving a pair of invariant laminations—and those universal circles play a key role in relating the dynamical structure to the geometry of M. In my minicourse and in my colloquium talk, I will introduce the idea of *zippers*, which give a new and direct way to construct universal circles, streamlining the known constructions in many cases, and giving a host of new constructions in others. In particular, zippers—and their associated universal circles—may be constructed directly from homological objects (uniform quasimorphisms), causal structures (uniform left orders), and many other structures.

If M is a hyperbolic 3-manifold fibering over the circle, then the fundamental group of M acts faithfully by homeomorphisms on a circle—the circle at infinity of the universal cover of the fiber—preserving a pair of invariant (stable and unstable) laminations. Many different kinds of dynamical structures including taut foliations and quasigeodesic or pseudo-Anosov flows are known to give rise to universal circles—a circle with a faithful action of the fundamental group preserving a pair of invariant laminations—and those universal circles play a key role in relating the dynamical structure to the geometry of M. In my minicourse and in my colloquium talk, I will introduce the idea of *zippers*, which give a new and direct way to construct universal circles, streamlining the known constructions in many cases, and giving a host of new constructions in others. In particular, zippers—and their associated universal circles—may be constructed directly from homological objects (uniform quasimorphisms), causal structures (uniform left orders), and many other structures.

If M is a hyperbolic 3-manifold fibering over the circle, then the fundamental group of M acts faithfully by homeomorphisms on a circle—the circle at infinity of the universal cover of the fiber—preserving a pair of invariant (stable and unstable) laminations. Many different kinds of dynamical structures including taut foliations and quasigeodesic or pseudo-Anosov flows are known to give rise to universal circles—a circle with a faithful action of the fundamental group preserving a pair of invariant laminations—and those universal circles play a key role in relating the dynamical structure to the geometry of M. In my minicourse and in my colloquium talk, I will introduce the idea of *zippers*, which give a new and direct way to construct universal circles, streamlining the known constructions in many cases, and giving a host of new constructions in others. In particular, zippers—and their associated universal circles—may be constructed directly from homological objects (uniform quasimorphisms), causal structures (uniform left orders), and many other structures.

A knot bounds an oriented compact connected surface in the 3-sphere, and consequently in the 4-ball. The 4-genus of a knot is the minimal genus among all such surfaces in the 4-ball, and the 4-genus of a link is defined analogously. In this talk, I will discuss lower bounds on the 4-genus derived from Cheeger-Gromov-von Neumann rho-invariants. This is joint work with Jae Choon Cha and Min Hoon Kim.