Gauge Theory and Topology Seminar: Cube tilings and alternating links
Gauge Theory and Topology Seminar, SEMINARS
May 5, 2023 3:30 pm - 4:30 pm
Science Center 507
Address: 1 Oxford Street, Cambridge, MA 02138 USA
Joshua Greene - Boston College
Consider a planar graph G, and form the lattice of integer-valued flows on G. Is it the case that this lattice embeds into the lattice of integer points in Euclidean space in such a way that each unit cube with integer vertices contains a point of the embedded sublattice?
Consider instead an alternating link L, and form the double-cover of the three-sphere branched along L. Is it the case that this space bounds a smooth four-manifold with trivial rational homology groups?
Under the correspondence that takes L to its Tait graph G, we conjecture that the answers to these two questions are the same. I will explain why a positive answer to the second implies a positive answer to the first using Floer homology. I will then explain why a positive answer to the first implies a positive answer to the second under the added hypothesis that each unit cube contains a *unique* point of the embedded sublattice.
This is joint work, the forward direction with Slaven Jabuka and the partial reverse direction with Brendan Owens.