Hi! I'm Ian Banfield

I am a Ph.D student in Mathematics at Boston College, working with Eli Grigsby.

I am interested in low-dimensional topology, knots and braids, and homological invariants thereof. The focus of my current work is to study connections between geometric properties of links and algebraic structures on the braid group.
CV | papers on the arxiv

email: ian.banfield@bc.edu | phone: 617-552-3878 | my office: Maloney Hall 538.
mailing address: Department of Mathematics, Maloney Hall, Boston College, Chestnut Hill, MA-02467.

my research.

My work is in low-dimensional topology, studying links and braids from geometric and algebraic viewpoints.

A strongly quasipositive braid.

I think about the interplay between geometric and algebraic invariants of knots and links and applications to low-dimensional topology and contact geometry. I use a variety of methods: classical 3-dimensional manifold topology such as sutured manifold theory, as well as modern homological methods and representation theory. At the moment, I am working on two related projects: the first is a study of a class of strongly quasipositive links, and the second is to understand the contact-geometric information in sutured Khovanov homology.

The first paper completed from these projects is my preprint Almost all strongly quasipositive braid closures are fibered. I use the Birman-Ko-Lee presentation of the braid group to show that strongly quasipositive braid closures satisfying a simple condition are fibered. It is quite surprising (and maybe even counterintuitive) that my argument implies that the probability that a strongly quasipositive braid is fibered approaches 1 if one fixes the strand index and increases the word length. I also prove a geometric interpretation of my condition. These braid closures have interesting applications to open questions about unknotting numbers, the slice-ribbon conjecture and L-spaces, and I plan to investigate these questions in the future.

The second project is motivated by the question whether Khovanov-homology can detect the contact structure induced by a fibered link. I have proved a number of partial answers for certain classes of braid closures and hope to finish writing these up towards the end of 2016. I have also defined a Khovanov-flavored invariant of links in thickened punctured discs and constructed a braid-invariant from this. The software section includes a calculator for this Khovanov homology theory.

my teaching.

Click on the course to download the syllabus. Recently, I have incorporated active learning techniques into my teaching, and moved away from tradional assessment methods to projects and group work, to great success. My full teaching portfolio and philosophy is available on request.

Calculus I

Calculus I

at BC: Fall 2013, Fall 2014, Fall 2015.

software.

Annular Khovanov Calculator

SuturedKhovanov

Calculates the sutured Khovanov homology of a braid closure, as an 𝔰𝔩₂ module. Runs on windows operating systems.

N-punctured Khovanov Calculator

NPuncturedKhovanov

This program calculates the triply graded Khovanov-type homology in thickened punctured disks I constructed.