All talks will be in Skiles 006. 

The conference will begin with one talk at 4:00 on Friday (preceded by a tea) and end by around 1:00 on Sunday.

Lightning talks session 1:

Lightning talks session 2:

Titles and Abstracts

Neda Bagherifard
Title: A 3-dimensional approach to the Andrews-Curtis Conjecture
Abstract: The Andrews-Curtis conjecture, proposed in the 1950s, asserts that all balanced presentations of the trivial group can be simplified through specific transformations known as stable Andrews-Curtis transformations to the trivial presentation. This conjecture is connected to the smooth 4-dimensional Poincaré conjecture, and some believe that both conjectures may be incorrect. In my talk, however, I will explain a partial connection between the Andrews-Curtis equivalence and a notion of equivalence on certain families of 3-manifolds, and I will demonstrate that a version of the conjecture is not true. This is joint work with Eaman Eftekhary.

Joe Breen
Title: Lagrangian sliceness and Legendrian derivatives
Abstract: A knot is Lagrangian slice if it bounds a Lagrangian disk in the symplectic 4-ball. One may ask for such a Lagrangian disk to be decomposable, or regular, in analogy to asking for a smooth slice disk to be ribbon, or handle-ribbon. We currently do not have a characterization of Lagrangian sliceness,or an understanding of whether the additional decomposability/regularity conditions are equivalent. In this talk, I will discuss recent work on the latter, and forthcoming joint work with A. Zupan making progress toward the former.

Hyunki Min
Title: Knot Floer homology, Sutured Floer homology and contact structures
Abstract: The rank of knot Floer homology contains a lot of information about a knot and its complement. Ghiggini and Ni showed that the rank of HFK in the top grading is 1 if and only if the knot is fibered. Ghiggini–Spano and Ni also showed that, for fibered knots, the rank of HFK in the second-to-top grading is equal to one more than the number of fixed points of the monodromy. There have been several attempts to estimate the rank of HFK in the second-to-top grading. Baldwin and Vela-Vick showed that the rank is greater than or equal to 1 if the knot is fibered, and Ni showed that the rank is greater than or equal to the rank in the top grading under certain additional conditions. In this talk, we give a new estimate of the rank of HFK for non-fibered knots in the second-to-top grading using sutured Floer homology and contact structures. This is joint work with Konstantinos Varvarezos.

Peter Ozsváth
Title: TBA
Abstract: TBA

Ina Petkova
Title: Spectral GRID invariants and Lagrangian cobordisms
Abstract: Knot Floer homology is a powerful invariant of knots and links, developed by Ozsvath and Szabo in the early 2000s. Among other properties, it detects the genus, detects fiberedness, and gives a lower bound to the 4-ball genus. The original definition involves counting homomorphic curves in a high-dimensional manifold, and as a result the invariant can be hard to compute. In 2007, Manolecu, Ozsvath, and Sarkar came up with a purely combinatorial description of knot Floer homology for knots in the 3-sphere, called grid homology. Soon after, Ozsvath, Szabo, and Thurston defined invariants of Legendrian knots using grid homology. We show that the filtered version of these GRID invariants, and consequently their associated invariants in a certain spectral sequence for grid homology, obstruct decomposable Lagrangian cobordisms in the symplectization of the standard contact structure, strengthening a result of Baldwin, Lidman, and Wong. This is joint work with Jubeir, Schwartz, Winkeler, and Wong.

Randy Van Why
Title: TBA
Abstract: TBA

Jonathan Zung
Title: Ziggurats and taut foliations
Abstract: If L is a link in a 3-manifold, which Dehn surgery multislopes give rise to 3-manifolds with taut foliations? In this talk, I will discuss the ziggurat phenomenon: if one restricts to foliations transverse to a fixed flow on the link complement, the set of multislopes typically has a fractal staircase shape with rational corners. In work in progress with Thomas Massoni, we explain the ziggurat phenomenon in some contexts using tools from contact geometry.

Lightning talk

Geoffrey Baring
Title: scl and quasimorphisms in the free group
Abstract: Stable commutator length is a stable word measure on groups which has been used to show various group theoretic results. In a paper of Danny Calegari, he proves that the scl can be computed on chains in the free group case by a rational linear programming problem which runs through certain surfaces and maximizes their Euler characteristic. On the dual side, Bavard Duality tells us that scl is equal to the supremum over homogeneous quasimorphisms. My current project with my adviser is to dualize the linear programming problem and reinterpret the variables as counting quasimorphisms to find an extremal quasimorphism for the given word.

Nilangshu Bhattacharyya
Title: Stable Homotopy Type for Planar Trivalent Graphs with Perfect Matchings
Abstract: In this talk, I will describe a space-level refinement of 2-factor homology by constructing a stable homotopy type associated to a distinguished family of planar trivalent graphs equipped with perfect matchings. The resulting spectrum is similar to the Lipshitz–Sarkar Khovanov spectrum for links and yields a stable homotopy invariant of planar trivalent graphs with perfect matchings. Moreover, we show that the closed webs obtained by performing flattenings at each crossing of an oriented link diagram—arising naturally in the context of sl3​ link homology—belong to this family of planar trivalent graphs for which the spectrum is built.

Fabio Capovilla-Searle
Title: Top degree cohomology of congruence subgroups of symplectic groups
Abstract: The cohomology of arithmetic groups has connections to many areas of mathematics such as number theory and diffeomorphism groups. Classifying spaces of congruence subgroups of symplectic groups have an algebro-geometric interpretation as the moduli space of principally polarized abelian varieties with level structures. These congruence subgroups Sp_2n(Z,L) are the kernel of the mod-L reduction map Sp_2n(Z) to Sp_2n(Z / L). By work of Borel-Serre, H^i(Sp_2n(Z / L)) vanishes for i > n^2. I will report on lower bounds in the top degree i = n^2. The key tools in the proof are the theory of Steinberg modules and highly connected simplicial complexes.

George Clare Kennedy
Title: Investigating Aptamer Space
Abstract: Aptamers are short strands of RNA which function as chemical antibodies, binding to specific target molecules. They are engineered using a process which produces many possible candidates, but only a few of these will have actually strong binding affinity; most of the rest are false positives. As aptamers are fundamentally genetic sequences, it is an intuitive hypothesis that aptamers which are genetically "close" should also have similar binding affinity, or at least perform similarly in the candidate-generating process. We can also compare structural differences between aptamers and posit a similar line of thinking. However, using traditional clustering techniques as well as the Mapper algorithm from topological data analysis, we show that neither of these distance measures appears effective at grouping together aptamers as they relate to binding affinity.

Matthew Elpers
Title: Characterizing slopes for more three manifolds
Abstract: For any homotopy class h in any compact orientable 3-manifold M which is closed or has exclusively torus boundary components, we produce infinitely many pairs of distinct knots representing h with orientation-preserving homeomorphic 0-surgeries.

Siavash Jafarizadeh
Title: Functoriality for $p$-equivariant Khovanov homology
Abstract: I will prove that equivariant khovanov homology for $p$-periodic links is functorial for special class of equivariant cobordisms. I will mention the steps needed for the proof of functoriality in the ordinary Khovanov homology and skew them in my favor.

Yangxiao Luo
Title: Cornered skein lasagna theory
Abstract: The Khovanov-Rozansky skein lasagna module was introduced by Morrison-Walker-Wedrich as an invariant of 4-manifold with a framed oriented link in the boundary. I will discuss an extension of the skein lasagna theory to 4-manifolds with codimension 2 corners, its behavior under gluing, and applications to trisections of 4-manifolds. This is joint work with Sarah Blackwell and Slava Krushkal.

Qingfeng Lyu
Title: Persistent foliarity of some (1,1) non-L-space knots
Abstract: We discuss an approach to finding taut foliations in (1,1) non-L-space knot complements. We use it to show that (1,1) almost L-space knots are persistently foliar.

Rob McConkey
Title: Theta, Traffic, and Connected-Sums
Abstract: Bar-Natan and van der Veen recently presented their new invariant for knots, theta. It is a powerful invariant which effectively distinguishes knots that is also easily computable. We will talk about the invariant and see how the invariant is represented. Talk about how cars driving on our knots can be used to help prove things for the invariant. And how we use the cars and the resulting traffic functions to show additivity of theta under the connected sum.

Sriram Raghunath
Title: Symplectic Annular Khovanov homology for symmetric knots
Abstract: When a knot diagram is symmetric, it induces an action on the Khovanov chain complex of the knot. We can analyze the equivariant cohomology of the chain complex with respect to this action to understand relationships between the Khovanov homology of the original knot and the Khovanov homology of the quotient knot. Seidel and Smith have defined a symplectic reformulation of combinatorial Khovanov homology, and they have used localization techniques in Floer theory to study the symplectic Khovanov homology of 2-periodic knots. In our work, we define an annular version of symplectic Khovanov homology, and apply this theory to investigate the symplectic Khovanov homology of 2-periodic and strongly invertible knots. This talk is based on joint work with Kristen Hendricks and Cheuk Yu Mak.

Nur Saglam
Title: Contact $\tau$-Invariant for dual knots
Abstract: The Heegaard Floer $\tau$-invariant is a knot concordance invariant that bounds the 4–ball genus of a knot in $S^{3}$ defined by Ozsváth and Szabó using Knot Floer Homology. Later, Raoux generalized their construction to knots in a rational homology sphere and obtained lower bounds in some of these invariants. There are various computations of $\tau$-Invariant for many manifolds like this, for example, contact $\tau$-invariant defined by Hedden. Motivated by these works, we aim to compute it for dual knots. Let L be a Legendrian knot in $S^{3}$ with the standard contact structure and $(Y, \xi)$ be the manifold obtained by Legendrian surgery on L. Our goal is to calculate the $\tau$-invariant $\tau_{\xi}(S^3_{tb-1}(K),K_{tb-1})$ of the dual knot K in terms of the smooth $\tau$-invariant $\tau(K)$ in $S^{3}$ and the surgery info. This is joint work with Katherine Raoux.

Luke Seaton
Title: Whitehead doubles and θ
Abstract: We introduce a computational tool to generate t-twisted, s-clasped Whitehead doubles of knots. Using this tool, we propose a conjecture describing the behavior of θ on untwisted Whitehead doubles and verify the conjecture for all prime knots with fewer than 13 crossings. Our results provide supporting evidence for a broader conjecture that θ, introduced by Bar-Natan and van der Veen, may be equal to the two-loop polynomial, a previously studied knot invariant.

Alexander Simons
Title: Homology of the Legendrian Contact DGA for -1 closures of positive braids.
Abstract: Given a Legendrian knot, Chekanov introduced a way to associate a differential graded algebra (DGA) to the knot. The homology of this DGA is an invariant that has been used to differentiate Legendrian knots that cannot be distinguished with tb and rotation numbers. In this brief talk, I will explain how to study the homology of these DGAs using noncommutative Groebner bases.

Ivan So
Title: Equivariant knots and real monopole
Abstract: Equivariant knots are knots with a specified symmetry. Extending from the definition of equivariant knots, there are the notions of equivariant Seifert surface and equivariant slice surface in D^4. However, even for the equivariant Seifert genus, it is hard to determine as there is no analogue of Seifert algorithm for equivariant knots. In this talk, I will talk about a work in progress which utilize invariants from real monopole Floer homotopy type to give a lower bound for equivariant slice genus. This is a joint work with Jin Miyazawa.

Maxwell Throm
Title: Symmetric knots and sl(3) homology
Abstract: An equivariant link is a link that has a finite order group action preserving the link. From this group action,   one can define a compatible group action on the Khovanov homology of the knot. This gives rank inequalities on the odd and even Khovanov homologies for the knot as in Stoffregen-Zhang. We hope to extend this to sl(3) link homology.

Rithwik Susheel Vidyarthi
Title: Knot Floer Homology and the Borromean Knot
Abstract: Mapping class groups of surfaces are interesting objects to study. One way to study them is to form the mapping torus, which is a three-manifold. For surfaces with a single boundary, this naturally gives rise to a 3-manifold along with a knot inside it, and we can assign it the Knot Floer complex. A natural question to ask is when is this injective? When the monodromy is the Identity, it is in fact injective, and this corresponds to the Borromean knot. We will try to answer this question when the monodromy is a boundary Dehn twist.

Kevin Yeh
Title: Train track folding automaton and the detection of T(2,3)#T(2,3) by knot Floer homology
Abstract: We aim to show that there are no hyperbolic knots with the same HFK as that of T(2,3)#T(2,3). This amounts to classifying all pseudo-Anosov maps on the disk with six marked points that lift to fixed-point free pseudo Anosov maps. To that end, we give a description of an algorithm that generates all such possible train tracks on the disk that can carry such maps. From these we can perform an extensive case analysis to enumerate all possible maps on the disk carried by these train tracks. We then rule out all of the cases using some calculations on the Alexander polynomial, or calculations on the homology in the lift.