| Name: | Sarah Pritchard |
|---|---|
| Webpage: | https://sites.google.com/view/sarahpritchard |
| Institution: | Georgia Institute of Technology, Atlanta GA |
| Project: | Relationships Between Combinatorial Knot Invariants |
| Mentor: | Dr. Kristen Hendricks |
About My Project
Topologists define a knot as a smooth embedding $S^1 \hookrightarrow S^3$: that is, a closed (potentially knotted) loop in 3-dimensional space. We can also consider links; a link is an embedding of a disjoint union $S^1 \cup S^1 \cup \cdots \cup S^1 \hookrightarrow S^3$, where each copy of $S^1$ is called a component. Mathematical knots are interesting objects of study because they appear in nature (DNA and protein folding, for example), can be quite complex, and are closely related to the study of manifolds in topology; knots form one case of the embedding problem, and a theorem of Lickorish and Wallace states that every closed 3-manifold can be described in terms of a link with an integer associated to each component. We can associate invariants to knots or links to help in distinguishing them. In this project, we will study the skew-filtered chain map $\iota_K$, which can be computed using information from the $CFK^\infty$ complex of a knot. Using this map, we can compute some involutive concordance invariants. These invariants provide 4-dimensional data on a knot; one can detect the fact that $4_1$ does not bound a smooth disk in $B^4$, for example, which is unusual for invariants like this. Previously, $\iota_K$ has been computed for torus knots, alternating knots, and (at a previous iteration of this REU) some pretzel knots. We hope to compute $\iota_K$ for some 10- and 11-crossing (1,1) knots for which $\iota_K$ has not been computed and try to understand when a (1,1) diagram gives us enough information to easily compute $\iota_K$.
Research Log
Week 1: May 25 - May 29
This week, we met our project mentor and have begun reading some background material for the project. Our reading included topics such as the Alexander polynomial and an introduction to homology; we’ve discussed chain complexes and chain maps and how these relate to the study of knots (for example, the chain complex $CFK^\infty(K)$). We briefly discussed (1,1) knots and their diagrams, but we'll learn more about them later.
Week 2: May 31 - June 4
This week, we presented our project goals to the rest of the REU participants. Then, we continued our background reading on concordance, the Sarkar involution, and some involutive concordance invariants.
Week 3: June 7 - June 11
This week, we continued learning some background information on topics in topology; this included Dehn surgery, Heegard decompositions, homology, and some background on Heegard Floer homology for 3-manifolds. Anna and I computed $\iota_k$ and the involutive concordance invariants $\underline{V}_0,\overline{V}_0,$ and $V_0$ for $10_{128}$ and $10_{132}$. Anna worked out a method for computing these for staircases in general and we discussed this together.
Week 4: June 14 - June 18
This week, we learned more about homology and cohomology. We also learned about (1,1) knots and (1,1) diagrams, which are a type of diagram representing these knots on the surface of the torus. We continued computing the map $\iota_k$ and the involutive concordance invariants $\underline{V}_0,\overline{V}_0,$ and $V_0$ for some 10-crossing knots.
Week 5: June 21 - June 25
This week, we learned more about involutive Heegard Floer homology and two types of knots that are of interest to us: L-space knots and thin knots. $\iota_k$ has been computed for some of these. We computed $\iota_k$ for some 10-crossing knots: a few together, and a few independently. While working on the computation for $10_{145}$ independently, I realized this computation was more complex than some of the others because of two nested boxes in the CFK complex. I set this computation aside.
Week 6: June 28 - July 2
In addition to computing $\iota_k$ (and some involutive concordance invariants) for some 10- and 11-crossing knots for which it is not known, we'd also like to learn more about (1,1) diagrams and what they can and cannot look like. This week, we found a diagonal arrow in a $s \neq 0$ diagram; it was not previously known that this could happen. In the process, we noticed some errors in the existing literature. We learned about another parametrization for (1,1) diagrams (the Rasmussen parametrization) so that we could use python to generate (1,1) diagrams for knots. Then, we created a table of the knots in this paper and checked for errors; we corrected two.
Week 7: July 6 - July 9
This week, we revisited the computation for $\iota_k$ of $10_{145}$. We created a python program to help with this computation and also discussed it as a group with Dr. Hendricks. Another question we'd like to answer is whether (1,1) diagrams for non-L-space $s=0$ knots can have an arrow of length 3. I identified two candidate knots and will check them for this.
Week 8: July 12 - July 16
This week, we completed the computations for $\iota_k$ and the involutive concordance invariants for eight of the 11-crossing knots. These computations went more quickly with the help of our python program, though there were still some things we needed to input and check manually. I continued thinking about the possibility of a length 3 arrow on an $s=0$ diagram.
Results so far:
Since the last week of the REU is approaching, here's what we've accomplished so far.
- Computed the map $\iota_k$ and the involutive concordance invariants $\underline{V}_0, \overline{V}_0,$ and $V_0$ for five 10-crossing knots and eight 11-crossing knots for which they were not previously known.
- Verified the Heegard diagrams for twenty-six 11- and 12-crossing knots in existing literature, identified three which were incorrect, and corrected two of them.
- Identified a (1,1) diagram whose $CFK$ complex has a diagonal arrow. (That is, a knot whose (1,1) diagram contains a bigon containing both points).