## Bio

Professor of Mathematics, University of Virginia.

Research interests: Low-dimensional geometry and topology

## Bio

Professor of Mathematics, University of Virginia.

Research interests: Low-dimensional geometry and topology

- 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.

PDF1.22 MB - 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, π

_{1}(G) injects into π_{1}(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 π_{1}(M) of rank 2.PDF3.14 MB - Freedman M, Krushkal V. Engel groups and universal surgery models. Journal of Topology. 2020;13:1302–1316.
We introduce a collection of 1/2-π

_{1}-null 4-dimensional surgery problems. This is an intermediate notion between the classically studied universal surgery models and the π_{1}-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.1707.07800.pdf1.41 MB - 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.

1901.05951.pdf1.37 MB - 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.structure_of_the_flow.pdf2.07 MB