My research group works in geometric knot theory, which is a field of study which relates the topological and geometric properties of knotted curves in space. We have two current projects:

- Understand the topology and geometry of random space curves. We model random curves by random polygons with a fixed number of edges. By understanding the differential, algebraic, and symplectic geometry of the manifold of n-gons, we are able to prove theorems about the probability distributions of geometric and topological quantities associated to the polygons. For instance, we were recently able to derive an exact formula for the expected total curvature of a space n-gon using an identification between the manifold of n-gons of length 2 and the Grassmann manifold of 2-spaces in complex n-space. We organized the 2013 Georgia Topology Conference on these topics, and a 2015 workshop at the Simons Center for Geometry and Physics.
- Understand the distribution of topology in random knot diagrams. We are working on a census of knot and link diagrams with the goal of gathering experimental evidence on questions like: what is the distribution of knot types among all 8 crossing knot diagrams?

The UGA group currently includes graduate students Tom Needham, and Harrison Chapman, undergraduates Eric Lybrand, Malik Henry, and Hollis Neel, and UGA faculty member Gary Iliev. We also work with Clayton Shonkwiler, Tetsuo Deguchi, Alexander Grosberg, Rob Kusner, Alessia Mandini, Erica Uehara, and Bertrand Duplantier.

## Past Research Group Members

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## Past Research Projects

### 2010-2011 — The Symplectic and Sub-Riemannian Geometry of Polygon Space

Jason Cantarella (faculty advisor), Graduate students: Jennifer Belton, Michael Berglund, Nick Castro, Whitney George, Sayonita Ghosh-Hajra, Joel Oakley, Thomas Needham, Matt Mastin, Maren Turbow. Undergraduates: Ellie Dannenberg, Jacob Rooney.

We are currently investigating the structure of the space of polygons in R^3 with fixed edge lengths. This is a fascinating topological space, usually a smooth manifold, with a variety of interesting structures on it. We began by investigating the sub-Riemannian structure defined by rotating pairs of adjacent edges around the line connecting their endpoints. We believe that the n-plane distribution structure defined for an n vertex polygon by these rotations is bracket-generating and we are working on proving it. In the process, we’ve found some intriguing connections between these flows and the symplectic structure on polygon space.

### 2008–2010 — Symmetries and Tabulation of Composite Links

Jason Cantarella (faculty advisor), Jason Parsley Graduate students: Ted Ashton, Yang Liu, Al LaPointe (spring 2007), Steve Lane, Matt Mastin, Whitney Montgomery, Laura Nunley, Aja Johnson, Amy Reeves, Gregory Schmidt, Jae-Ho Shin, Joe Tenini. Undergraduate students: Jacob Rooney, Daniel Cellucci, James Dabbs, Ellie Dannenberg, Alex Moore, Emmanuel Obi (spring 2007), Meredith Perrie, Rachel Whitaker (fall 2006).

The research group is currently studying the question of intrinsic symmetries for links. Given a link of two components, the question usually comes down to whether there is a motion of the curves in the link which interchanges the two components while preserving the orientations of each component. An example of such a motion is shown on the main page.

We are currently tabulating the symmetries of all links of 8 and fewer crossings. This table is the key ingredient in constructing a table of composite links. Once we complete our tabulation of the symmetries, we hope to construct the first definitive table of composite knots and links with 10 and fewer crossings. During the 2009/2010 academic year, Whitney Montgomery, Matt Mastin and Jason Cantarella are writing up our results on the symmetries of links with 8 and fewer crossings, assisted by Ellie Dannenberg and Jacob Rooney (undergraduates), who are computing symmetries as needed. We submitted the paper describing these symmetries in Fall 2010.

Jason Cantarella and Ellie Dannenberg (undergraduate) gave a talk How much (subatomic) rope does it take to tie a knot? to the Franklin College Dean’s Council in fall 2009.

Ellie Dannenberg (undergraduate) gave a poster Intrinsic Symmetries of Links at the Joint AMS-MAA-SIAM meetings in San Francisco in January 2010, and the talk Symmetries of Knots and Links at the 2009 UnKnot Conference at Dennison University.

Meredith Perrie (undergraduate) gave the talk *Computing the ropelength of tight links* at the 2007 Mercer University Undergraduate Research in Mathematics Conference (MUURMaC).

James Dabbs (undergraduate) gave a talk at the 2007 Young Mathematicians Conference at Ohio State

### 2007 Summer – Geometric Flows for Plane Curves

Jason Cantarella (faculty advisor), Graduate Students: Matt Mastin, Aja Johnson Undergraduate Students: Ricky Biggs, Erik Forseth, Chris Green, Paul James, Drew Lupton, Jesse Rao, Matt Wise.

This group worked on the behavior of energies for plane curves under geometric evolution equations, studying results of Pavel Exner, Michael Gage, Gerhard Huisken, Chad Mullikan, and others. The essential questions in this area are simple, but hard to analyze. For instance, suppose that a plane curve is varied so as to decrease its length as fast as possible. What is the limiting shape of the curve? Will the curve ever develop a self-intersection, if it does not have self intersections to start? Will the curve ever get rid any self-intersections present at the start of the flow?

Some of these questions were settled in the 1980’s, but many fascinating questions remain wide open.

The group held daily meetings over an 8 week period, eventually resolving into a few working groups. Paul James worked with Ricky Biggs on computer models of the curves and their flow. Drew Lupton and Jesse Rao got interested in the question of finding curves with bounded distortion. Erik Forseth completed an ambitious project of reinterpreting the results of the important papers of Michael Gage for the curve-shortening flow to deal with a flow that continuously rescaled the curve to remain at constant length. Chris Green proved a theorem about the distortion of polygonal knots, improving a result of Chad Mullikan. And Matt Wise worked on a theorem about energy functionals for knotted curves in space. Ricky Biggs deserves special credit for helping with several of the working groups. In addition, Biggs worked with Jesse Rao on an approach to finding curves of given length and maximum average L^p chord length. Biggs and Rao derived a criticality condition for these curves. Biggs also analyzed preprints of Exner et. al., finding an important correction which leads to a restatement of the main results.

- Erik Forseth’s writeup about length-rescaled curvature flow
- Chris Green’s theorem on distortion for polygonal knots
- Matt Wise’s conjecture on existence of critical curves for knot energies.

Matt Wise (undergraduate) gave a talk at the Mid-Hudson Math Conference.

### 2005/2006 — Ropelength of Composite Knots

Jason Cantarella (faculty advisor), Jason Parsley (postdoc), Graduate Students: Ted Ashton, Adam Fletcher, Aja Johnson, Chad Mullikin, Amy Reeves, Undergraduate Students: John Foreman, Rachel Whitaker.

The physicist Thomas Kephart at Vanderbilt has pointed out in a neat paper that certain subatomic particles called glueballs might be well-modelled by tightly knotted tubes. Working with Kephart, the group, together with Eric Rawdon worked on finding a table of lengths for tight knots and links to match with glueball data. A key problem is that the best-known knots are the so-called prime knots but glueballs are modeled just as well by composite knots .

Shortest lengths for many prime knots and links have been posted by several authors, but nobody had tabulated ropelengths for composite knots and links in a systematic way. This group spent the year working on a program to automatically generate all composite knots and links up to 12 crossings. The knots are being tightened at the Vanderbilt University ACCRE computer cluster.

Rachel Whitaker (undergraduate) gave the talk *Generating composite knots* at the 2006 UGA CURO symposium .

Rachel Whitaker (undergraduate) gave the talk *Connected sums of knots* at MathFest 2006, and was one of only 8 students to win an *MAA Prize for Student Talks*!

### 2004/2005 — Individualized Research Projects

Jason Cantarella (faculty advisor), Graduate students: Ted Ashton and Chad Mullikin, Undergraduate students: Seth Dowling, John Foreman, Michael Piatek, Amy Reeves.

During this year, students worked on individual projects. Ashton spent most of his time tuning the octrope system and writing the arXiv preprint Self-contact Sets for 50 Tightly Knotted and Linked Tubes . Mullikin worked on a series of computations of knots minimizing Gromov’s distortion, which were later described in his Ph.D. thesis . Dowling worked on automatically rendering grayscale images with dot stippling (much like the rendering program Piranesi ). Reeves worked on understanding configuration spaces of polygons. Foreman worked on a pinscreen animation project which was also supported by Ideas for Creative Exploration .

Ted Ashton (graduate student) gave the talk *Computing the curvature of polygons* at the 2005 Spring Southeastern Section AMS Meeting.

John Foreman (undergraduate) gave the talk *Pinscreen Animation* at the ICE Experimental Animation Night.

### Summer 2004 — Mathematics and Art

Jason Cantarella (faculty advisor), Rusty Wallace (faculty advisor), Chad Mullikin (graduate TA), Cody Van der Kaay (graduate TA), Undergraduate students: Cathy Covington, Kit Hughes, Julie Orlemanski, Ryan Grady, Alessa Ellefson, Seth Dowling, Michael Piatek.

During this summer, Cantarella and Piatek completed the Tsnnls linear algebra solver, described in the preprint TSNNLS: A solver for large sparse least squares problems with non-negative variables ( arXiv:cs.MS/0408029 ). They used this to create the first version of the RidgeRunner knot tightening software, which was described in the paper Visualizing the tightening of knots, which was presented at the computer graphics conference IEEE VIS’05 . Some of the animations generated by ridgerunner appeared in the *Mathematical Art Exhibit* at the 2006 AMS-MAA Annual Meetings.

### 2003/2004 — Searching for stuck unknots

Jason Cantarella (faculty advisor), Graduate students: Ted Ashton, Mike Beck, Casey Bowman, Chad Mullikin, Undergraduate students: John Foreman, John Gibbs, Amy Reeves.

If we think of the set of open space polygons with fixed edgelengths, we can see that some are embedded (no self-intersections) while others are singular (pairs of edges intersect each other). Do the singular polygons divide the space of polygons into different components? Or can any such polygon be reconfigured to any other without self-intersections? A few examples of such polygons were provided by Cantarella and Johnston and Cocan and O’Rourke (several other authors, including Clark and Venema and Aloupis, Ewald, and Toussaint also made important contributions to this area). This group worked on a computer search algorithm for finding more examples of such polygons in an organized way.

In the spring semester, Ashton and Cantarella completed the octrope library for finding the self-contacts of a tube in space. The library became the subject of the paper A fast octree-based algorithm for computing ropelength, which appeared in the volume *Physical and Numerical Models in Knot Theory and their Application to the Life Sciences* in 2005.

Ted Ashton (graduate student) gave the talk *A fast octree-based algorithm for computing ropelength* at the 2004 Spring Southeastern Section AMS Meeting.

Michael Piatek (undergraduate) gave the talk *Implementing the gradient of ropelength* at the 2004 Spring Southeastern Section AMS meeting.

John Forman (undergraduate) gave the talk *Geometric representations of nth hulls of knotted curves* at the 2004 AMS-MAA Joint Meetings in Phoenix, AZ.

### Summer 2003 — Massively Parallel Computation of Geometrically Optimal Knots

Jason Cantarella (faculty advisor), Chad Mullikin (graduate TA), Monica Shaw (graduate TA), Undergraduate Students: Allison Diana, Igor Ganichev, John Foreman, John Gibbs, Michael Piatek, Amy Reeves.

This group worked on software for tightening simulated knotted tubes in parallel on many machines. The code produced by Piatek laid the foundations for the ridgerunner project (described later), while Shaw and Diana worked on a prototype version of octrope.

Amy Reeves (undergraduate) gave the talk *Massively parallel computation of optimal knots: I* at MathFest 2003.

Michael Piatek (undergraduate) gave the talk * Massively parallel computation of optimal knots: II* at MathFest 2003.

### 2002/2003 — The Knotted Pearl Problem

Jason Cantarella (faculty advisor), Joseph Fu (faculty advisor), Nancy Wrinkle (postdoc), Graduate students: Ted Ashton, Jim Blair, Tanya Cofer, Kenny Little, Chad Mullikin, Jenn Robinson, Undergraduate students: John Foreman, John Gibbs, Amy Reeves. <br />

This group split into two parts. The subgroup of Fu, Wrinkle, and Ashton worked on describing the structure of the simple clasp formed when two ropes are pulled across each other. This work later became part of the paper Criticality for the Gehring Link Problem, which will appear in Geometry and Topology in 2006. The rest of the group worked on the 15 pearls conjecture of Oshiro and Maehara , which states that it requires 15 tangent spheres to form a nontrivial knot. This problem turned out to be quite difficult.

### 2001/2002 — The Ropelength of Knots

Jason Cantarella (faculty advisor), Nancy Wrinkle (postdoc), Graduate students: Xander Faber, Chad Mullikin, Wade Schueneman, Heunggi Park, Undergraduate students: Darren Wolford, Qixing Zhang.

This group considered the relationship between the minimum length of unit-diameter rope required to tie a knot (its *ropelength*) and the minimum crossing number of that knot. Eventually, Cantarella, Faber, and Mullikin were able to prove that ropelength is bounded above by a constant multiple of the square of the crossing number. This result appeared in 2003 in Topology and it’s Applications as the paper Upper Bounds for Ropelength as a function of Crossing Number. Eventually, Yuanan Diao was able to improve this result and show that ropelength was bounded by the 3/2 power of crossing number.