| Name: | Matthew Issac |
|---|---|
| Email: | matthewissac (at) rutgers (dot) edu |
| Home Institution: | Rutgers University - New Brunswick |
| Project: | Algebraic Invariants of Pretzel Knots |
About My Project
Heegaard Floer Homology allows us to associate a knot to a doubly filtered chain complex. Maps between such complexes are called chain maps. We are interested in an involutive chain map, iota_k, and its computation for pretzel knots P(-2,m,n) with m and n odd. This map has been computed for a torus knots and we hope our computation will help provide more examples.
Research Log
Week 1
I was given a brief introduction to knot theory, knot invariants, and some basic homological algebra. Some basic knot invariants I learned about are: three colorability, Alexander polynomial, and Heegaard Floer Homology. I was given a crash course in computing homologies and had plenty of examples to practice with. Moreover, I learned how the Euler Characteristic generalized to bi-graded homologies and how this information encompasses the Alexander polynomial. Finally, I capped the week off by looking into an introductory paper to Heegaard Floer Homology.
Week 2
We were introduced to filtered chain complexes and chain maps. Later on, we saw the Sarkar Involution an involutive, filtered, chain map. This is a crucial piece of data for iota_k, since iota_k^2 is chain homotopic to the Sarkar involution. We also worked through an example of a computation of iota_k for the figure 8 knot.Week 3
We saw the technical side of CFK^∞, the chain complex we have been working with. We also saw how the generators of this complex come from intersection points on Heegaard diagrams.Week 4
We have started to read Knot Floer Homology of (1,1)-knots by Goda, Matsuda, and Morifuji. They have already computed CFK^∞ for P(-2,m,n) with m and n odd. We started to understand their notation and worked out what CFK^∞ is for P(-2,5,5) and worked out iota_k for this knot.Week 5
I began working on a calculator which would accept integers n and m and compute the shape of the complex. There were initally many bugs but I eventually ironed it out and we can now compute more examples.Week 6
I have begun thinking about a change of basis to do to the complex and how to create a box and staircase pattern.Week 7
I am beginning to understand how the indices relate to one another, in particular I found certain conditions on when two generators lie in the same filtration level.Week 8
We have begun discussing the main theorem, in addition to new invariants. We have drafted this and are now preparing our final presentation.Week 9
Finished our presentation but still more work to be done. We still need to prove main theorem and finish the final change of basis.References & Links
Epilogue
After the program finished, we proved our main theorem. All that remains is the paper, which is currently being written. Here are the papers I have read for my project:- Knot Floer homology of (1,1)-knots. , Goda, Matsuda, and Morifuji - ..
- An introduction to Heegaard Floer homology , Ozsvath and Szabo - Retrieved from here..
- My Mentor's Website
- The REU Website
HTML 4 font rendering:
∂y/∂t = ∂y/∂x √2 =1.414
If f(t)= ∫t 1 dx/x then
f(t) → ∞ as t → 0. This really means:
(∀ε ∈ℝ, ε>0) (∃δ>0)
f(δ) > 1/ε .
ℕ (natural numbers), ℤ (integers), ℚ (rationals),
ℝ (reals), ℂ (complexes)