Research

2021

Akhmechet R, Johnson P, Krushkal V. Lattice cohomology and  q-series invariants of 3-manifolds. arXiv:2109.14139. 2021.

An invariant is introduced for negative definite plumbed 3-manifolds equipped with a spinc-structure. It unifies and extends two theories with rather different origins and structures. One theory is lattice cohomology, motivated by the study of normal surface singularities, known to be isomorphic to the Heegaard Floer homology for certain classes of plumbed 3-manifolds. Another specialization  gives BPS q-series which satisfy some remarkable modularity properties and recover SU(2) quantum invariants of 3-manifolds at roots of unity. 
In particular, our work gives rise to a 2-variable refinement of the \(\widehat Z\)-invariant. 

Given a finite CW complex $K$, we use a version of the Goodwillie-Weiss tower to formulate an obstruction theory for embeddings of $K$ into a Euclidean space $\R^d$. For $2$-dimensional complexes in $\R^4$, a geometric analogue is also introduced, based on intersections of Whitney disks and more generally on the intersection theory of Whitney towers developed by Schneiderman and Teichner. The focus in this paper is on the first obstruction beyond the classical embedding obstruction of van Kampen. In this case  we show the two approaches lead to essentially the same obstruction. We also relate it to the Arnold class in the cohomology of configuration spaces. The obstructions are shown to be realized in a family of examples. Conjectures are formulated, relating higher versions of these homotopy-theoretic, geometric and cohomological theories.

Akhmechet R, Krushkal V, Willis M. Stable homotopy refinement of quantum annular homology. Compos. Math. 157 (2021), 710-769. 2021.

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.

2020

Freedman M, Krushkal V. Filling links and spines in 3-manifolds, with an appendix by C. Leininger and A. Reid. arXiv:2010.15644; to appear in Communications in Analysis and Geometry. 2020.

We introduce and study the notion of filling links in 3-manifolds: a link L is filling in M if for any 1-spine G of M which is disjoint from L, π1(G) injects into π1(M∖L). A weaker "k-filling" version concerns injectivity modulo k-th term of the lower central series. For each k>1 we construct a k-filling link in the 3-torus. The proof relies on an extension of the Stallings theorem which may be of independent interest. We discuss notions related to "filling" links in 3-manifolds, and formulate several open problems. The appendix by C. Leininger and A. Reid establishes the existence of a filling hyperbolic link in any closed orientable 3-manifold with π1(M) of rank 2.

Akhmechet R, Krushkal V, Willis M. Towards an sl2 action on the annular Khovanov spectrum. https://arxiv.org/abs/2011.11234. 2020.

Given a link in the thickened annulus, its annular Khovanov homology carries an action of the Lie algebra sl2, which is natural with respect to annular link cobordisms. We consider the problem of lifting this action to the stable homotopy refinement of the annular homology. As part of this program, the actions of the standard generators of sl2 are lifted to maps of spectra. The main new technical ingredients developed in this paper, which may be of independent interest, concern certain types of cancellations in the cube of resolutions and the resulting more intricate structure of the moduli spaces in the framed flow category.

Freedman M, Krushkal V. Engel groups and universal surgery models. Journal of Topology. 2020;13:1302-1316.

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.

2019

Freedman M, Krushkal V. Universal Surgery Problems with Trivial Lagrangian. Math. Res. Lett. 2019;26:1587-1601.

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.

Agol I, Krushkal V. Structure of the flow and Yamada polynomials of cubic graphs. Breadth in contemporary topology, Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, RI. 2019;102:1–20.

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.