# Research

## Symplectic and Contact Geometry and Topology:

Below are some of the most recent research paper by our group:

#### Local rigidity, contact homeomorphisms, and conformal factors (arXiv:2001.08729)

Author: Michael Usher

In this paper, Mike shows that if the image of a Legendrian submanifold under a contact homeomorphism (i.e. a homeomorphism that is a C0-limit of contactomorphisms) is smooth then it is Legendrian, assuming only positive local lower bounds on the conformal factors of the approximating contactomorphisms. More generally the analogous result holds for coisotropic submanifolds in the sense of arXiv:1306.6367

#### Symplectic trisections and the adjunction inequality (arXiv:2009.11263)

Author: Peter Lambert-Cole

In this paper, Peter establishs a version of the adjunction inequality for closed symplectic 4-manifolds. As in a previous paper on the Thom conjecture (arxiv:1807.10131), he uses contact geometry and trisections of 4-manifolds to reduce this inequality to the slice-Bennequin inequality for knots in the 4-ball. As this latter result can be proved using Khovanov homology, he completely avoid gauge theoretic techniques. This inequality can be used to give gauge-theory-free proofs of several landmark results in 4-manifold topology, such as detecting exotic smooth structures, the symplectic Thom conjecture, and exluding connected sum decompositions of certain symplectic 4-manifolds.

#### Bordered Floer homology and contact structures (arXiv:2011.08672 )

Authors: Akram Alishahi, Viktória Földvári, Kristen Hendricks, Joan Licata, Ina Petkova, Vera Vértesi

In this paper, they introduce a contact invariant in the bordered sutured Heegaard Floer homology of a contact three-manifold with boundary. They use a special class of foliated open books to construct admissible bordered sutured Heegaard diagrams and identify well-defined classes cD and cA in the corresponding bordered sutured modules. Foliated open books exhibit user-friendly gluing behavior, and we show that the pairing on invariants induced by gluing compatible foliated open books recovers the Heegaard Floer contact invariant for closed contact manifolds. They also consider a natural map associated to forgetting the foliation F in favor of the dividing set, and show that it maps the bordered sutured invariant to the contact invariant of a sutured manifold defined by Honda-Kazez-Matić.

#### Symplectic (−2)-spheres and the symplectomorphism group of small rational 4-manifolds, II (arXiv:1911.11073 )

Authors:  Jun Li, Tian-Jun Li, Weiwei Wu

For (CP^2#5\overline{CP^2},ω), let N_ω be the number of (−2)-symplectic spherical homology classes. In this paper,  they completely determine theTorelli symplectic mapping class group (Torelli SMCG): the Torelli SMCG is trivial if N_ω>8; it is π_0(Diff^+(S^2,5)) if N_ω=0 (by Paul Seidel and Jonathan Evans); it is π_0(Diff+(S^2,4)) in the remaining case.

#### Filtering the Heegaard Floer contact invariant (arXiv:1603.02673 )

Authors: Cagatay Kutluhan, Gordana Matic, Jeremy Van Horn-Morris, Andy Wand

In this paper, they define an invariant of contact structures in dimension three from Heegaard Floer homology. This invariant takes values in the set Z^{≥0}∪{∞}. It is zero for overtwisted contact structures, ∞ for Stein fillable contact structures, non-decreasing under Legendrian surgery, and computable from any supporting open book decomposition. As an application, they

obstruct Stein fillability on contact 3-manifolds with non-vanishing Ozsváth-Szabó contact class.

## Low Dimensional Topology:

#### Random graph embeddings with general edge potentials (arXiv:2205.09049)

Authors: Jason Cantarella, Tetsuo Deguchi, Clayton Shonkwiler, Erica Uehara

In this paper, they study random embeddings of polymer networks distributed according to any potential energy which can be expressed in terms of distances between pairs of monomers. This includes freely jointed chains, steric effects, Lennard-Jones potentials, bending energies, and other physically realistic models. A configuration of n monomers in R^d can be written as a collection of d coordinate vectors, each in R^n. The first main result of this paper is that entries from different coordinate vectors are uncorrelated, even when they are different coordinates of the same monomer. They predict that this property holds in realistic simulations and in actual polymer configurations (in the absence of an external field). The second main contribution is a theorem explaining when and how a probability distribution on embeddings of a complicated graph may be pushed forward to a distribution on embeddings of a simpler graph to aid in computations. This construction is based on the idea of chain maps in homology theory. They use it to give a new formula for edge covariances in phantom network theory and to compute some expectations for a freely-jointed network.

#### Diffeomorphisms of the 4-sphere, Cerf theory and Montesinos twins (arXiv:2102.12890)

Author: David Gay

One way to better understand the smooth mapping class group of the 4-sphere would be to give a list of generators in the form of explicit diffeomorphisms supported in neighborhoods of submanifolds, in analogy with Dehn twists on surfaces. As a step in this direction, Dave describe a surjective homomorphism from a group associated to loops of 2-spheres in S^2×S^2's onto this smooth mapping class group, discuss two natural and in some sense complementary subgroups of the domain of this homomomorphism, show that one is in the kernel, and give generators as above for the image of the other. These generators are described as twists along Montesinos twins, i.e. pairs of embedded 2-spheres in the 4-sphere intersecting transversely at two points.

#### Trisecting 4-manifolds (arXiv:1205.1565)

Authors: David T. Gay, Robion Kirby

In this paper, they show that any smooth, closed, oriented, connected 4--manifold can be trisected into three copies of ♮^k(S^1×B^3), intersecting pairwise in 3--dimensional handlebodies, with triple intersection a closed 2--dimensional surface. Such a trisection is unique up to a natural stabilization operation. This is analogous to the existence, and uniqueness up to stabilization, of Heegaard splittings of 3--manifolds. A trisection of a 4--manifold X arises from a Morse 2--function G:X→B^2 and the obvious trisection of B^2, in much the same way that a Heegaard splitting of a 3--manifold Y arises from a Morse function g:Y→B^1 and the obvious bisection of B^1.

#### Upsilon invariant for graphs and the homology cobordism group of homology cylinders (arXiv:2202.10614)

Author: Akram Alishahi

Upsilon is a homomorphism on the smooth concordance group of knots defined by Ozsváth, Stipsicz and Szabó. In this paper, Akram defines a generalization of upsilon for a family of embedded graphs in rational homolog spheres. This generalized invariant induces a homomorphism on the homology cobordism group of homology cylinders. To define this invariant, she uses tangle Floer homology, lifts relative gradings on tangle Floer homology to absolute gradings (for certain tangles) and proves a concatenation formula for it.