Topology

Twist-roll spun knots are a well-studied family of knotted 2-spheres in the 4-sphere. The inputs for determining a twist roll spun knots are an integer number of twists, a number of rolls, and a choice of classical knot in the 3-sphere. I will explain why adding twists, while changing the…

Knot invariants are typically used to give a negative answer to the question of when two embeddings are ambiently isotopic, and rarely to give a positive answer. An exception is the celebrated result of Freedman and Quinn that if the complement of a 2-sphere embedded in the 4-sphere has…

Let S be a finite set of invertible 2-by-2 matrices with algebraic entries. Is there an algorithm to determine a presentation for <S>? We shall pose this question, and suggest a possible avenue to answering it. Note that every finite-volume hyperbolic 3-manifold arises as above. Beyond…

Given n points and a smooth Jordan curve in the complex plane, what is the minimum degree of a non-constant polynomial which maps all of the points to the curve?  It is easy to bound the degree above by n-1, while if the points are collinear and the curve is an ellipse, then the degree is…

If Δ is a contractible compact d-manifold, then its boundary Σ will be a homology (d-1)-sphere, but the boundary need not be simply connected and Δ need not be homeomorphic to the d-disk.  In joint work with Randal-Williams, we show that the topological group consisting of homeomorphisms of…

In the 1970's dihedral representations of knot groups were used to define twisted signature-type invariants which generalize the older invariants of Levine and Tristram. The most prominent examples are the Casson-Gordon invariants, which provide obstructions to being topologically slice as well…

Yang--Mills gauge theory with gauge group SU(2) has played a significant role in the study of the topology of 3- and 4-manifolds. However, there is not much known about applications of gauge theory with other gauge groups in the study of low dimensional manifolds. In this talk, I will discuss a…

Suppose you take a 1 x L strip of paper, twist it around in space, and tape the (short) ends together to make a paper Moebius band. In this talk I'll prove that you must have L > sqrt(3) and also that there is a unique limit that emerges if you have examples with L tending to sqrt(3). B.…

Knot Floer homology is a powerful link invariant. In its most general version, it is a bigraded chain complex over a polynomial ring F[U,V]. In this talk, I will describe the structure theorem of such objects - they are a direct sum of snake complexes and local systems - and explain what…

Curve graphs are crucial tools for studying mapping class groups of surfaces. However, many basic questions on their geometry remain open. In this talk, we will shed light on the geometry of curve graphs by describing “filtrations” of them by hyperbolic graphs. These filtrations yield quasi-…