Title: The fractional Dehn twist coefficients of braids

Description: Project is to (1) search for an upper bound and a lower bound of the fractional Dehn twist coefficient (FDTC) of an n-braid where n is greater than or equal to 5, using the Birman-Ko-Lee left canonical form of braids, (2) develop an algorithm to compute the FDTC, and (3) study braids whose FDTC increases after braid stabilization. 

Preferred Background: Basic knot theory (eg. Rolfsen’s book) and having coding experiences.

Project leaders:

• Keiko Kawamuro (University  of Iowa)
• Margaret Doig (Creighton University)

Title: Khovanov homology and symmetric unions

 
Description: Symmetric unions, introduced by Kinoshita and Terasaka [KT57], provide a geometric framework for generating ribbon knots. A symmetric union is a knot that admits a symmetric union diagram, constructed as follows: place a knot diagram \(D\) and its mirror image −D symmetrically on either side of an axis in the plane and take their connected sum, D # (−D), across this axis. Then, insert vertical twists (tangles consisting of a specific integer number of half-twists) along the axis of symmetry. For a symmetric union knot K the minimum number of twist regions needed to construct a symmetric union diagram for K is denoted by tw(K).

While it is known that every symmetric union is a ribbon knot, the converse remains an open problem. The goal of this project is to study symmetric union knots using immersed curve invariants for 4-ended tangles in Khovanov homology [KWZ24]. For example, if a knot K admits a symmetric union diagram constructed from a knot diagram D, then is the rank of Khovanov homology of K greater than or equal to the rank of the Khovanov homology of D or D # (−D)? Can one get any bounds on tw(K) from Khovanov homology?

References:

[KT57] Shin’ichi Kinoshita and Hidetaka Terasaka. On unions of knots. Osaka 

Mathematical Journal, 9(2): 131-153, 1957.

[KWZ24] Artem Kotelskiy, Liam Watson, and Claudius Zibrowius. Immersed 

Preferred Background: General knowledge of low-dimensional topology (e.g., knot theory, 3-manifolds), familiarity with some tools from Heegaard Floer homology; familiarity with the immersed curves picture specifically is a plus but not a requirement!

Project leaders:

• Akram Alishahi (University of Georgia)
•  Kristen Hendricks (Rutgers)

Title: Fukaya-Seidel categories

Description: Mirror symmetry is a string-theory inspired duality between a symplectic manifold X and a candidate mirror complex manifold Y. Homological mirror symmetry (HMS) is a conjectural equivalence between categorical invariants: the Fukaya category of X and the bounded derived category of coherent sheaves on Y. HMS was initially studied between compact Calabi-Yau manifolds and more recently between Fano manifolds and their non-compact, exact mirrors. In many general type examples, the mirror is not closed and non-exact. These mirrors are often Landau-Ginzburg (LG) models.

This project aims to understand the A-infinity structure of the Fukaya-Seidel categories appearing in our work from the previous WiSCon workshop. We start with studying known cases of the cotangent fiber, Fano hypersurfaces of projective space, other compact as well as non-compact examples and the partially wrapped Fukaya category with stops.

Preferred Background: Familiarity with Lagrangian intersection Floer theory and wrapped Fukaya categories or Fukaya-Seidel categories.

Project leaders:

• Haniya Azam (LUMS, Pakistan)
• Catherine Cannizzo (Columbia)

Title: Hypersurfaces of generalized complex manifolds

Description: Generalized complex manifolds, initially introduced by Hitchin, are generalizations of both complex and symplectic structures. This is a proposal to study hypersurfaces of generalized complex manifolds. Izu Vaisman extended the fact that an oriented hypersurface of an almost Hermitian manifold inherits a metric almost contact structure to the generalized complex case. But, when delving into the literature it becomes clear that there is a lack of non-classical examples. One of the goals of this project is to explore induced structures on hypersurfaces of generalized complex manifolds and to construct interesting examples.

Preferred Background: The background expectations are very basic knowledge about Hermitian manifolds, Kahler structures, almost contact structures, and  generalized complex manifolds.

Project leaders:

• Aissa Wade (Pennsylvania State University)
• Honglei Lang (Chinese Agricultural University)

Title: Symplectic Geometry of Abelian Polygon Spaces in ℝ³

Description: The abelian polygon space is the space of chains in ℝ³ with fixed side lengths that terminate at a specific plane z = α, modulo rotations around the z-axis. It is a symplectic manifold with a natural symplectic form obtained by symplectic reduction. We plan to determine which of  these spaces are symplectic monotone, identifying the corresponding side lengths and diffeomorphism types and Poincaré polynomials, as well as the (reflexive) moment polytopes for a natural toric action. Moreover, we would like to obtain a description of these spaces as quiver varieties through abelianization of the star-shaped quivers associated to the moduli spaces of spatial polygons.

Preferred Background: The members of the group would preferably be familiar with Hamiltonian actions on symplectic manifolds and symplectic reduction

Project leaders:

• Leonor Godinho (Instituto Superior Técnico, Portugal)
• Alessia Mandini (University of Verona, Italy)

Title: Contact manifolds, fibered knots and Floer theory

Description: Giroux’s correspondence allows us to describe contact manifolds via open book decompositions and fibered knots. Through this description we have connections to Heegaard Floer theory (and the contact invariant) and mapping class groups (via the monodromy of the fiber surface). Our goal will be to study invariants of contact manifolds coming from these different descriptions to investigate questions about tight and overtwisted contact structures.

Preferred Background: Familiarity with contact structures on 3-manifolds and Heegaard Floer theoretic invariants.

Project leaders:

• Katherine Raoux (University of Arkansas)
• Linh Truong (University of Michigan)