Anna Antal's DIMACS REU Webpage

Name: Anna Antal
Email: anna.antal@rutgers.edu
Home Institution: Rutgers University-New Brunswick
Most Recent Project: Relationships between combinatorial knot invariants

About My Project

Abstract: In this project we worked with involutive Heegaard Floer homology, developed by Hendricks and Manolescu as a refinement to Heegaard Floer homology. Involutive Heegaard Floer homology introduced two new knot concordance invariants, \(\underline{V}_0\) and \(\overline{V}_0\). These invariants are interesting, because unlike other concordance invariants like \(\tau, \epsilon\), and \(\nu\), they do not necessarily vanish on knots of finite concordance order, like the figure-eight knot. We computed \(\underline{V}_0\) and \(\overline{V}_0\) for all non-thin and non-\(L\)-space 10 and 11-crossing (1,1)-knots for which they were not yet known. We also studied (1,1)-diagrams, and the relationships between them and the resulting \(CFK^{\infty}\) chain complexes.

Our Team

Weekly Summaries

Week 1

This week was an exciting start to the program! It was wonderful to see all the participants and hear about their projects in the orientation meeting. Our group met every day this week. Dr. Hendricks gave us an overview of our project and lectured us on necessary background information. We covered the Alexander polynomail, basic homological algebra, bigraded chain complexes, filtered complexes, and mirrors and tensor products. Sarah and I were assigned small exercises related to each of the topics, which I worked on throughout the week. In the second half of the week we began working on our first presentation, which is an explanation of our research topic and goals.

Week 2

We began the week by preparing for, and then giving our introductory project presentation to our fellow participants. It was interesting to hear the variety of projects that everyone will be working on. The rest of my week was filled with lots of new material, and lots of work on the assigned exercises. In our daily group meetings we covered the Sarkar involution, knot concordance, and the involutive concordance invariants. We solidified are goals for the duration of the summer, and I gained a solid understanding of our research questions. I spent most of my time learning the new material and computing the solutions to the exercises.

Week 3

Week 3 was filled with learning more context, solving exercises, and working out some small original results! In our daily group meetings we spent time discussing the solutions to exercises, and Dr. Hendricks lectured us on knot surgery, Heegaard decompositions, and the connection between Heegaard Floer decompositions of 3-manifolds and surgery. At the start of the week Sarah and I also met with Karuna, who talked through some of the exercises with us. Sarah and I worked on computing \(\iota_K\) and the involutive concordance invariants for the 10-crossing knots \(10_{128}\) and \(10_{132}\). I also worked on figuring out these computations for a general staircase.

Week 4

This week the focus in the lectures was on homology and cohomology. We discussed definitions, worked through some examples, like computing the homology and cohomology of \(S^n\), \(S^1 \times S^1\), and \(\mathbb{R}\mathbb{P}^2\). We also used the Mayer-Vietoris exact sequence to compute the homology of a \(g\)-holed surface and the homology of \(p\)-surgery on the unknot \(S_p^3(U)\). We began working on computing the homology of \(p\)-surgery on an arbitrary knot. Next to our studies of homology and cohomology, Sarah and I continued our work of finding \(\iota_K\) and the involutive concordance invariants for the remainging 10-crossing \((1,1)\) knots. I did the computations for \(10_{136}\) and \(10_{139}\), and we started the computation for \(10_{145}\) together.

Week 5

We began this week by continuing our study of homology, and with some comprehensive lectures on the broader context of our project. Specifically, we worked through calculating the homology of a \(p/q\) surgergy on an arbitrary knot. Professor Hendricks explained the history of Involutive Heegaard Floer Homology, and all the work that led to its study. It gave me a much better understanding of what we are working on this summer, and why it is interesting. I spent the second half of the week working on some computations. I crafted the Heegaard diagrams for some 11 and 12-crossing knots, and found the \(CFK^{\infty}\) complex for \(12n_{591}\) from its Heegaard diagram. I also began working on the question of whether a (1,1)-diagram of a knot \(K\) can yield a diagonal arrow in \(CFK^{\infty}(K)\).

Week 6

It was an exciting week filled with surprising discoveries. In my search for a diagonal arrow in a knot's \(CFK^{\infty}\) complex, I found a small mistake in the literature. The four integers \((k,r,c,s)\) that should determine the Heegaard diagram for the knot \(12n_{749}\) were given incorrectly in a paper. The correct result is in Knot Polynomials and Knot Homologies by Rasmussen. By adding to python code provided by Professor Hendricks, I checked the rest of the data in this section of the paper, found two other mistakes, and corrected one of them. I also began computing \(\iota(K)\) for the knot \(11n_{12}\).

Week 7

I spent the first couple days of the week finishing the computation of \(\iota_K\) for \(11n_{12}\). I also found the involutive concordance invariants for this knot. I spent the rest of the week writing a program in Python that will help speed up future computations of \(\iota_K\), especially for more complicated knots. The code reduces the number of possiblities for \(\iota_K\), and so significantly decreases the amount of calculations that need to be done by hand. In our group meeting, Dr. Hendricks gave us examples of where results from research projects like ours are being used in current research.

Week 8

I spent the penultimate week of the program calculating \(\iota_K\) and the concordance invariants for as many of the remaining 11-crossing knots as possible. I did these calculations for \(11n_{38}, 11n_{57}\), and \(11n_{61}\), and their mirrors. The program I wrote in Python was very useful in reducing computation time. In addition, I drew the Heegaard diagrams for \(11n_{96}\) and \(11n_{135}\), and used them to find the \(CFK^{\infty}\) complexes for each of the two knots.

Week 9

The last week was spend finishing all our planned calculations, preparing and presenting our final project presentation, and writing our final paper. I found the Heegaard diagram for the last 11-crossing knot of interest, \(11n_{111}\), and used it to get the \(CFK^{\infty}\) complex. I also calculated \(\iota_K\) and the concordance invariants for this knot and the knots \(11n_{20}\) and \(11n_{135}\). Working on this project was a very rewarding experience, and I am very grateful for the opportunity!

Some Presentations

References & Links

Here are some relevant resources:
  1. Knot Floer Homology of (1,1)-Knots, Goda, Matsuda, Morifuji. - link.springer.com
  2. Involutive Heegaard Floer Homology, Hendricks, Manolescu. - arXiv.org
  3. An Introduction to Knot Floer Homology, Manolescu. - arXiv.org
  4. Grid Homology for Knots and Links, Ozsvath, Stipsicz, Szabo.
  5. Geometry of (1,1)-Knots and Knot Floer Homology, Racz. - dataspace.princeton.edu
  6. Knot Polynomials and Knot Homolgies, Rasmussen. - arXiv.org

Funding

This project was supported by NSF CAREER grant DMS-2019396.

About My Project

We are examining mutations of polynomials by studying cluster algebras and toric varieties.

Abstract: Mutations arise in mirror symmetry and so-called cluster theory when there are different natural coordinate systems on the same solution set. For example, consider the set of 2-dimensional subspaces of a 4-dimensional vector space. Taking two basis vectors we get a \(2 \times 4\) matrix, and any two of the determinants of \(2 \times 2\) matrices inside the resulting \(2 \times 4\) matrix give coordinates for the set. The various coordinates systems are related by a mutation. We will explore this and other examples. More concretely a mutation of a polynomial \(f(x,y)\) in two variables \(x,y\) with factor \(h(x,y)\) is one obtained by replacing each monomial \(x^i y^j\) with \(x^i y^j h(x,y)^{l(i,j)}\) where \(l(i,j)\) is some power that depends in a linear way on the powers \(i,j\). For example, a mutation is given by replacing \(y\) with \(y(1+x)^{-1}\) and multiplying by \((1+x)^{-2}\). This mutation applied to \(f(x,y)\) results in the polynomial \(f'(x,y) = (1+x) + 2y + y^2\). If two polynomials are related by mutations we say they are mutation-equivalent. Problem: classify mutation-equivalence-classes of polynomials. There is a conjecture of Petracci and collaborators about what the mutation-equivalence classes are. We will investigate a version of this conjecture.

Our Team:

Weekly Summary

Week 1

It was an exciting first week of the program! I dived into the resources provided by my mentors, and have started to build a picture of the problems that we will be studying. I have become more familiar with what cluster algebras are, and have also started learning some algebraic geometry. During our group meetings we discussed some background material and potential questions to focus on throughout the summer.

Week 2

It was a very busy week! At the start of the week we gave our project introductory presentation and had the chance to listen to everyone else's as well. I also gave a presentation within my group about some algebraic geometry necessary for the project. Throughout the week I focused on studying toric varieties and basic algebraic geometry. I have a better idea now of the math necessary to approach our research questions. I am looking forward to continuing learning more new math!

Week 3

This week I started out with studying some more background material on cluster algebras. Then, I focused mainly on better understanding our main questions concerning mutations of polygons. I began with working with the definition of 0-mutable, and started writing a Macaulay2 program that applies mutations to polynomials. I would like to see all the mutations that can be applied to a given polynomial. In our group meetings we discussed the Grassmanian and other algebraic geometry. I will be giving a presentation about mutations of polygons next week.

Week 4

Another week filled with polynomials, polygons, and algebraic geometry! I began this week studying mutations of polygons, and how they relate to mutations of polynomials. This helped my understanding of our research questions. I gave a presentation overviewing the topic, and worked through some examples on my own. There are a lot of interesting questions to be investigated in this area. Later in the week I also did some more research on algebraic geometry. I'd like to better understand the connection between varieties and mutations of polynomials.

Week 5

I spent most of this week learning more albegraic geometry. I focused on the definition of the genus of a curve. This required knowing about projective space, the Hilbert function, and the Hilbert polynomial. At the end of the week we had a group meeting, where I presented what I was able to put together throughout the week. We also talked more about 0-mutable polynomials, and whether we can use code to determine whether a given polynomial is 0-mutable. For now, we are focusing on a specific trianlge, and are trying to classify all the polynomials that have this triangle as their convex hull.

Week 6

This week I did a lot of reading about mutations of polynomials and mutations of polygons. Because there are so many sources with different definitions of mutation, I started putting together a document of all the different defintions and their comparisons. There was also a key paper called Mirror Symmetry and smoothing Gorenstein toric affine 3-folds in this area published this week. We began looking at this paper, and trying to understand the provided examples. We have needed to slightly revise the definition of mutation we are working with.

Week 7

I split my time this week between learning some algebraic geometry and studying 0-mutable polynomials. I spent the first half of the week reading about morphisms. The two texts I found most helpful were Eisenbud's Commutative Algebra with a View Toward Algebraic Geometry and Kollar's The structure of algebraic threefolds: an introduction to Mori's program. After giving my presentation on morphisms hapfway through the week, I switched my focus back to the newly published paper Mirror Symmetry and smoothing Gorenstein toric affine 3-folds. I focused on working through all the details of the examples and figuring out the possible mutation data for the given polynomials.

Week 8

As we are nearing final presentations and final papers, I spent the week thinking about some small questions related to mutations of polynomials and how they relate to mutations of polygons. I worked on showing the equivalence of definitions of mutations in different papers, so that I could apply propositions from each paper.

Week 9

The week was a great conclusion to the program! I spent the first half of the week putting together the final project presentation with my partner Sam. Wednesday morning we had a successful presentation, and then had the opportunity to watch everyone else's presentations the rest of the week. The second half of the week we also focused on finishing the final paper. It was a nice feeling to see our summer's work put nicely together. Thank you to everyone who made this experience possible!

Some Presentations

References & Links

Here are some relevant resources:
  1. Introduction to Cluter Algebras, Fomin et al. - arXiv.org.
  2. Cluster Algebras: An Introduction, Williams - arXiv.org.
  3. Mirror Symmetry and smoothing Gorenstein toric affine 3-folds, Corti, Filip, Petracci - arXiv.org
  4. Mirror Symmetry and the Classification of Orbifold del Pezzo Surfaces, Akhtar, et al. - arXiv.org
  5. The structure of algebraic threefolds: an introduction to Mori's program, Kollar - Bulletin of the American Mathematical Society

Funding

This project was supported by the Rutgers University Mathematics Department.