Steven Sivek

A bordered Chekanov-Eliashberg algebra

Given a front projection of a Legendrian knot K in R 3 , which has been cut into several pieces along vertical lines, we assign a differential graded algebra to each piece and prove a van Kampen theorem describing the Chekanov–Eliashberg invariant of K as a pushout of these algebras. We then use this theorem to construct maps between the invariants of Legendrian knots related by certain tangle replacements, and to describe the linearized contact homology of Legendrian Whitehead doubles. Other consequences include a Mayer–Vietoris sequence for linearized contact homology and a van Kampen theorem for the characteristic algebra of a Legendrian knot.

Obstructions to Lagrangian concordance

We investigate the question of the existence of a Lagrangian concordance between two Legendrian knots in

The contact homology of Legendrian knots with maximal Thurston-Bennequin invariant

\mathbb{R}^3

Monopole Floer homology and Legendrian knots

. In particular, we give obstructions to a concordance from an arbitrary knot to the standard Legendrian unknot, in terms of normal rulings. We also place strong restrictions on knots that have concordances both to and from the unknot and construct an infinite family of knots with non-reversible concordances from the unknot. Finally, we use our obstructions to present a complete list of knots with up to 14 crossings that have Legendrian representatives that are Lagrangian slice.

Naturality in sutured monopole and instanton homology

We show that there exists a Legendrian knot with maximal Thurston-Bennequin invariant whose contact homology is trivial. We also provide another Legendrian knot which has the same knot type and classical invariants but nonvanishing contact homology.

Instanton Floer homology and contact structures

We use monopole Floer homology for sutured manifolds to construct invariants of unoriented Legendrian knots in a contact 3‐manifold. These invariants assign to a knot K Y elements of the monopole knot homology KHM. Y;K/, and they strongly resemble the knot Floer homology invariants of Lisca, Ozsvath, Stipsicz, and Szabo. We prove several vanishing results, investigate their behavior under contact surgeries, and use this to construct many examples of nonloose knots in overtwisted 3‐manifolds. We also show that these invariants are functorial with respect to Lagrangian concordance. 57M27, 57R58; 57R17

Contact structures and reducible surgeries

Kronheimer and Mrowka defined invariants of balanced sutured manifolds using monopole and instanton Floer homology. Their invariants assign isomorphism classes of modules to balanced sutured manifolds. In this paper, we introduce refinements of these invariants which assign much richer algebraic objects called projectively transitive systems of modules to balanced sutured manifolds and isomorphisms of such systems to diffeomorphisms of balanced sutured manifolds. Our work provides the foundation for extending these sutured Floer theories to other interesting functorial frameworks as well, and can be used to construct new invariants of contact structures and (perhaps) of knots and bordered 3-manifolds.

Fillings of unit cotangent bundles

We define an invariant of contact 3-manifolds with convex boundary using Kronheimer and Mrowka’s sutured instanton Floer homology theory. This is the first invariant of contact manifolds—with or without boundary—defined in the instanton Floer setting. We prove that our invariant vanishes for overtwisted contact structures and is nonzero for contact manifolds with boundary which embed into Stein fillable contact manifolds. Moreover, we propose a strategy by which our contact invariant might be used to relate the fundamental group of a closed contact 3-manifold to properties of its Stein fillings. Our construction is inspired by a reformulation of a similar invariant in the monopole Floer setting defined by Baldwin and Sivek (arXiv:1403.1930, 2014).

Quasi-alternating links with small determinant

We apply results from both contact topology and exceptional surgery theory to study when Legendrian surgery on a knot yields a reducible manifold. As an application, we show that a reducible surgery on a non-cabled positive knot of genus g must have slope 2g-1, leading to a proof of the cabling conjecture for positive knots of genus 2. Our techniques also produce bounds on the maximum Thurston-Bennequin numbers of cables.

Donaldson invariants of symplectic manifolds

We study the topology of exact and Stein fillings of the canonical contact structure on the unit cotangent bundle of a closed surface