3:30–4:30 pm
Eckhart Hall, Room 206 5734 S. University Ave. Chicago, IL 60637
Knotted handlebodies
Abstract:
Often, interesting knotting vanishes when allowed one extra dimension, e.g. knotted circles in 3-space all become isotopic when included into 4-space. Hughes, Kim and I recently found a new counterexample to this principle: for \(g>1\), there exists a pair of 3-dimensional genus-\(g\) solids in the 4-sphere with the same boundary, and that are homeomorphic relative to their boundary, but do not become isotopic rel boundary even when their interiors are pushed into the 5-dimensional ball. This proves a conjecture of Budney and Gabai (who previously constructed 3-balls in the 4-sphere with the same boundary that are not isotopic rel boundary) for \(g>1\) in a very strong sense.
In this talk, I’ll describe some motivation from 3-dimensional topology and useful/weird facts about higher-dimensional knots (e.g. knotted surfaces in 4-manifolds), show how to construct interesting codimension-2 knotting in dimensions 4 and 5 (joint with Mark Hughes and Seungwon Kim), and talk about related open problems.
