General Information
Project Description
A knot is a smooth (equivalently piecewise-linear) embedding of the circle S^1 into the three-sphere S^3.
A fundamental operation of producing new knots out of existing ones is called the satellite operation, which utilizes a pattern knot P inside a solid torus and a companion knot K. To construct the satellite knot P(K), we glue in the solid torus containing the pattern knot into a regular neighborhood of the companion knot K using a diffeomorphism of the torus boundaries.
One such set of satellite knots comes from the
Mazur Pattern.
In general, we may study knots up to a notion of equivalence (weaker than the standard knot equivalence in ambient isotopies), called concordance. Two knots are concordant if they jointly form the boundary of an annulus in S^3 x [0,1]. The development of Heegaard Floer knot Homology by P. Ozsvath and Z.Szabo led to the definition of an invariant of a concordance class of a knot, called the tau-invariant.
In
2016, A. Levine utilized bordered Floer homology, which is well adapted to studying satellite knots, to give a formula to calculate the tau-invariant of satellite knots with Mazur patterns.
In this project, we aim to generalize Levine's work by considering more general Mazur patterns. These patterns come from the (1,1)-patterns, which are patterns which can be drawn on the surface of a torus. The goal of the project is to provide explicit formulas for the tau-invariant of different types of (1,1)-Mazur patterns.
Weekly Log
- Week 1:
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We had orientation, and got to meet the other people in the program! In terms of content, our advisors gave us background lectures on our topic.
Some of the things we learned about include knots and different invariants, concordance, homology, bigraded complexes, Z and Z \oplus Z filtered chain complexes, bordered Heegaard diagrams, and the CFK,HFK,CFA-, and CFD complexes/modules.
We spent the week working on small exercises related to these topics, and preparing our presentation.
- Week 2:
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We spent the week computing tau invariants of simple knots like the trefoil and figure eight, making sure we could draw the Heegaard diagrams and work with the chain complexes. In particular, we replicated Lemma 4.3 and the CFA A_inf "m" relations given in remark 4.2 of Hom's Paper by finding disks from the bordered Heegaard diagrams for cable knots.
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Next, we replicated Levine's results for the computation of the tau-invariant for the Mazur pattern (section 4 of this paper), as this is the base case which we will work with to generalize. What's cool about these computations is that you only need to consider the tensor products of the CFD generators in the unstable chain!
- Week 3:
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We had lectures on the Alexander polynomial and Dehn Surgery, given by two of Prof. Hendricks' PhD students. It was pretty cool to learn about, and it helped to contextualize the role of Heegaard Floer homology in knot theory in general. For example, the Euler characteristic of HFK^{hat}(K) is the Alexander polynomial, which provides a genus bound on the knot!
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The Mazur pattern can be described as the following: go around the torus twice clockwise, then turn around, and go once counterclockwise, before crossing under the CCW strand and twice over the CW strands to join back at the start.
We constructed Heegaard diagrams for the modified Mazur pattern with n CCW rotations, with the connecting strand still crossing over these new rotations. We then worked through Adam's computation but for the A_inf modules we obtained from finding holomorphic disks from the bordered Heegaard diagrams from the new patterns, and came up with a formula for tau for this class of patterns.
- Week 4:
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We extended our results from the week before to compute tau for even more general patterns with m CCW rotations followed by m CW rotations. We also produced an algorithm to construct bordered Heegaard diagrams for Q_{m,n} inductively starting with Q_{1,1}. This is pretty exciting, but not all the details have been fleshed out about this yet. This would mean that we're done with tau, though. An easy corollary of this is the formula for tau when the winding number is +-1, which happens when |m-n| = +- 1.
- Week 5:
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We wrote up and filled in the details of our proof of the Q_{m,n} computation by finding all the m relations for the A_{infty} module and using a linear algebra argument to prove why the other relations don't matter. We also tried to generalize our results to accommodate r-framings of the satellite knot. The issue is that an r framed satellite operator doesn't necessarily send the unknot to an unknot, so it doesn't give us the nice homomorphism property that we'd like. This also means that our computation gets more complicated, as we can't find parts that would be isomorphic to the tensor product with an unknot exterior, which we used previously to help us recover the Alexander gradings for the generators in the tensor complex.
- Week 6:
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We did a deeper dive into the structure of A_{infty} modules as well as dg algebras to gain a better insight into the CFA and CFD modules. We also began work on writing a program to compute epsilon by working with arcslide decompositions and two-bridge link representations of our generalized Mazur patterns, which we found a braid decomposition for. By computing epsilon, we hope to find a family of counterexamples to an open question posed by Kirby about satellite operators with winding number +-1. The Mazur pattern is the first such example of a -1 winding number satellite operator which creates a satellite knot that is never exotically slice. We hope that our computation of epsilon for certain Q_{m,n} patterns may lead to more such examples, potentially allowing us to conjecture a family of patterns with this property.
- Week 7:
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We did more work into writing code to parse the DD and D modules that come from the composition of arcslides which give us our satellite knot. We worked towards getting Adam's results for the (2,1) and (2,-1) cable of the Mazur pattern (pattern knot is the cable knot, companion knot is the Mazur pattern here). We wrote code to turn the CFD complex into CFA complexes, and then tensored the CFA complex of the cabled Mazur pattern with the CFD complex of arbitrary knot exteriors to recover Adam's computation for the epsilon invariant.
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We also went to a low dimensional topology conference in Princeton on Thursday, which was very fun! It was really insightful to see what topics low dimensional topologists care about, and what some of the tools they use (a lot of Heegaard Floer when it comes to knot theory).
- Week 8:
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We worked on making our final presentation (and presented!), so a lot of figures and examples! We also continued to work on our epsilon computation, but ran into some roadblocks when trying to compute it for general Mazur patterns, which prompted another exploration of the arcslides and what they represent, as well as the order of which we should be composing our results. For some odd reason, tensoring T2 with T2^{-1} gave us the identity bimodule, but T^2{-1} with T2 gave us multiple generators, which was a big issue that we had to fix.
- Week 9:
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We wrote our final paper and reflection, and made many figures and polished our paper. Our work for epsilon stalled due to some discrepancy on the geometric intuition behind the braid decompositions for the bridge links we found in week 6. It seems like we have more work to do on that end, which we'll explore once the REU is over. Besides that, we found a potential corollary for nonisotopy and nonconcordance between a certain class of our generalized patterns, so we started this by finding the Schubert normal form for our patterns in terms of m and n. More work needs to be done on this end too... hopefully we will be able to wrap our loose ends up as we continue to work on this even after the REU is completed.
Presentations
Important Papers and References
Acknowledgements
My work was supported by NSF CAREER Grant DMS-2019396. I would like to thank my mentors for their guidance, Lazaros and Caleb for organizing the program, and the DIMACS REU for providing me with this opportunity to do research.
