Joshua Evan Greene

The slice-ribbon conjecture for 3-stranded pretzel knots

We determine the (smooth) concordance order of the 3-stranded pretzel knots

A spanning tree model for the Heegaard Floer homology of a branched double-cover

P(p, q, r)

Lattices, graphs, and Conway mutation

with

Chromatic capacities of graphs and hypergraphs

p, q, r

Strong Heegaard diagrams and strong L–spaces

odd. We show that each one of finite order is, in fact, ribbon, thereby proving the slice-ribbon conjecture for this family of knots. As corollaries we give new proofs of results first obtained by Fintushel-Stern and Casson-Gordon.

A note on applications of the d-invariant and Donaldson's theorem

Given a diagram of a link K in S^3, we write down a Heegaard diagram for the branched-double cover Sigma(K). The generators of the associated Heegaard Floer chain complex correspond to Kauffman states of the link diagram. Using this model we make some computations of the homology \hat{HF}(Sigma(K)) as a graded group. We also conjecture the existence of a delta-grading on \hat{HF}(Sigma(K)) analogous to the delta-grading on knot Floer and Khovanov homology.

The lens space realization problem

The d-invariant of an integral, positive definite lattice Λ records the minimal norm of a characteristic covector in each equivalence class

A New Short Proof of Kneser's Conjecture

({\textup{mod} \;}2\varLambda)

Homologically thin, non-quasi-alternating links

. We prove that the 2-isomorphism type of a connected graph is determined by the d-invariant of its lattice of integral flows (or cuts). As an application, we prove that a reduced, alternating link diagram is determined up to mutation by the Heegaard Floer homology of the link’s branched double-cover. Thus, alternating links with homeomorphic branched double-covers are mutants.

L-space surgeries, genus bounds, and the cabling conjecture

Abstract Given a hypergraph H , the chromatic capacity χ cap ( H ) of H is the largest k for which there exists a k -coloring of the edges of H such that, for every coloring of the vertices of H with the edge colors, there exists an edge that has the same color as all its vertices. We prove that if G is a graph on n vertices with chromatic number χ and chromatic capacity χ cap , then χ cap >(1− o (1)) χ/ 2n ln χ , extending a result of Brightwell and Kohayakawa. We also answer a question of Archer by constructing, for all r and χ , r -uniform hypergraphs attaining the bound χ cap = χ −1. Finally, we show that a connected graph G has χ cap ( G )=1 if and only if it is almost bipartite. In proving this result, we also obtain a structural characterization of such graphs in terms of forbidden subgraphs.