We construct a stable homotopy refinement of quantum annular homology, a link homology theory introduced by Beliakova, Putyra and Wehrli. For each r>2 we associate to an annular link L a naive Z/rZ-equivariant spectrum whose cohomology is isomorphic to the quantum annular homology of L as modules over Z[Z/rZ]. The construction relies on an equivariant version of the Burnside category approach of Lawson, Lipshitz and Sarkar. The quotient under the cyclic group action is shown to recover the stable homotopy refinement of annular Khovanov homology. We study spectrum level lifts of structural properties of quantum annular homology.

We introduce a collection of 1/2-π₁-null 4-dimensional surgery problems. This is an intermediate notion between the classically studied universal surgery models and the π₁-null kernels which are known to admit a solution in the topological category. Using geometric applications of the group-theoretic 2-Engel relation, we show that the 1/2-π1-null surgery problems are universal, in the sense that solving them is equivalent to establishing 4-dimensional topological surgery for all fundamental groups. As another application of these methods, we formulate a weaker version of the π1-null disk lemma and show that it is sufficient for proofs of topological surgery and s-cobordism theorems for good groups.

We establish a relation between the trace evaluation in SO(3) topological quantum field theory and evaluations of a topological Tutte polynomial. As an application, a generalization of the Tutte golden identity is proved for graphs on the torus.

We study the effect of Nielsen moves and their geometric counterparts, handle slides, on good boundary links. A collection of links, universal for 4-dimensional surgery, is shown to admit Seifert surfaces with a trivial Lagrangian. They are good boundary links, with Seifert matrices of a more general form than in known constructions of slice links. We show that a certain more restrictive condition on Seifert matrices is sufficient for proving the links are slice. We also give a correction of a Kirby calculus identity in [FK2], useful for constructing surgery kernels associated to link-slice problems.

A subset of R^d is called ``sticky'' if it cannot be isotoped off of itself by a small ambient isotopy. Sticky wild Cantor sets are constructed in R^d for each $d\geq 4$.

We establish a quadratic identity for the Yamada polynomial of ribbon cubic graphs in 3-space, extending the Tutte golden identity for planar cubic graphs. An application is given to the structure of the flow polynomial of cubic graphs at zero. The golden identity for the flow polynomial is conjectured to characterize planarity of cubic graphs, and we prove this conjecture for a certain infinite family of non-planar graphs. 
Further, we establish exponential growth of the number of chromatic polynomials of planar triangulations, answering a question of D. Treumann and E. Zaslow. The structure underlying these results is the chromatic algebra, and more generally the SO(3) topological quantum field theory.